On Supermembrane in D=4, multipleM0–brane in D=11 and Supersymmetric

Higher Spin Theories

Carlos Meliveo Garcıa

Ph.D. ThesisDepartment of Theoretical Physics and History of Science

University of the Basque Country (UPV/EHU)Leioa, December 2013

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Department of Theoretical Physics and History of Science

On Supermembrane in D=4, multipleM0–brane in D=11 and Supersymmetric

Higher Spin Theories

Supervised by Professor Igor A. Bandosfrom the University of the Basque Country and Ikerbasque Foundation

Submitted by Carlos Meliveo Garcıafor the degree of Doctor of Physics

A mi familia.

ACKNOWLEDGEMENTS

This thesis is based on the research carried out at the Department of TheoreticalPhysics and History of Science at the University of the Basque Country in the period2009-2013.

During these years, I have had the chance to meet many people who have helped me inmany different ways towards the completion of this thesis. For that reason, I would like toexpress my deepest gratitude to each of them.

First of all, I want to thank the person who has really made this thesis possible, mysupervisor Igor A. Bandos. Thanks Igor for the patience and support during these years.

I thank all members of the Department for their support and pleasant working atmo-sphere: Juan M. Aguirregabiria, Montserrat Barrio, Jose Juan Blanco, MariamBouhmadi, David Brizuela, Tom Broadhurst, Alberto Chamorro, Alexander Fein-stein, Luis Herrera, Jesus Ibanez, Ruth Lazcoz, Michele Modugno, Martın Rivas,Gunar Schnell, Vincenzo Salzano, Jose M. Martın Senovilla, Geza Toth, Jon Ur-restilla, Manuel A. Valle, Lianao Wu and Zoltan Zimboras. Specially, I would like tothank Raul Vera because without him my computer never worked properly and I thankInigo L. Egusquiza for encouraging me during the hard times.

On the other hand, I would like to express my gratitude to the Ph.D. students andPostdocs who have accompanied me along this thesis. I would like to mention Kepa Sousa,Irene Sendra, Diego Saez, Charlotte Van Hulse, Inigo Urizar, Giussepe Vitagliano,Borja Reina, Ariadna Montiel, Lluc Garcıa, Asier Lopez, Joanes Lizarraga, IagobaApellaniz and Pablo Jimeno, thank you very much, you have became really good friends.

I would also like to thank the support from my whole family; Ama, Aita, Bego, Sergio,Nadia including my nephews Ane, Jon and Markel and my old friends; Alvaro, Bolu,Cubas, Depra, Javi, Kraig, Mako and Ote.

Finally I want to thank Amaia for her patience, support and making this life easier.

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INTRODUCCION Y RESUMEN

El modelo estandar de la fısica de partıculas elementales ha demostrado tener la habilidadde describir una multitud de fenomenos que tienen lugar en la naturaleza, siendoampliamente respaldado por experimentos realizados en aceleradores de partıculas

que confirman sus predicciones. El ejemplo mas reciente es el anuncio del descubrimientodel boson de Higgs por parte de las colaboraciones ATLAS [1] y CMS [2].

El modelo estandar esta basado en la simetrıa gauge SU(3)×SU(2)×U(1) [3, 4] dondeSU(2)× U(1) es la simetrıa de la teorıa de campos gauge electrodebiles W±

µ , Zµ, Aµ queprovee la descripcion unificada de la interaccion electromagnetica y la debil. Fue propuestapor Sheldon L. Glashow [5], Steven Weinberg [6] y Abdus Salam [7] 1.

La simetrıa de gauge SU(2)× U(1) esta espontaneamente rota hasta un subgrupo U(1)”diagonal” de SU(2)× U(1). Como resultado, los bosones W±

µ , Zµ, que corresponden alos generadores de las simetrıas rotas, obtienen una masa gracias al mecanismo de Englert–Brout–Guralnik–Hagen–Kibble–Higgs [8–11]. En cambio, el campo electromagnetico Aµ secorresponde al generador de la simetrıa preservada y, entonces, describe una partıcula sinmasa, el foton.

La unificacion de las interacciones electromagneticas y debiles no es la primera en la historiade la fısica. Recordemos que en el siglo XIX James C. Maxwell demostro que fenomenoselectricos y magneticos, que anteriormente eran considerados como no relacionados, podıanser descritos por un conjunto unico de ecuaciones que a dıa de hoy llevan su nombre.Estas ecuaciones describen la teorıa del electromagnetismo y realizan la unificacion de losfenomenos electricos y magneticos [12].

La idea de unificacion puede ser interpretada como un camino hacia un entendimiento masprofundo de la naturaleza, incluso como un principio filosofico de unidad de la naturaleza:la naturaleza es unica y el entendimiento mas profundo puede –y tiene que– revelar laexplicacion comun de fenomenos aparentemente diferentes.

1Galardonados con el Premio Nobel en 1979.

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Ademas de teorıa de interacciones electrodebiles, el modelo estandar contiene la teorıade campos gauge Arµ, r = 1, ..., 8 del grupo SU(3), que se conoce como CromodinamicaCuantica (QCD por sus siglas en ingles). Los cuantos de este campo, que describe lasinteracciones fuertes, se llaman ”gluones”.

En el modelo estandar la QCD esta aparte de la intereccion electrodebil, incluso laconstante de acoplo de los gluones es uno de los parametros independientes. Eso se reflejaen la aparicion del grupo SU(3) en la simetrıa gauge del modelo estandar en productodirecto con SU(2)× U(1) y, en la falta de un mecanismo de mezcla con este ultimo. (Porotro lado, en la teorıa electrodebil la mezcla de SU(2) con U(1) aparece como el resultadode una rotura espontanea de simetrıa con preservacion del subgrupo U(1) diagonal deSU(2)× U(1).)

Ya en los anos 70 se empezo la busqueda de la teorıa que unificara mas estas interaccionestomando como base teorıas de campos gauge de grupos semisimples que contengan losgrupos SU(3), SU(2) y U(1) como subgrupos. Entre estas teorıas, que se conocen comoTeorıas de Gran Unificacion (GUT por sus siglas en ingles), las mas investigadas son las quetienen grupos de simetrıa gauge SU(5), SO(10) y E6.

Todas las GUT tienen en comun ciertas propiedades, entre ellas:

• La existencia de una unica constante de acoplo, en vez de las tres del modelo estandar:la separacion de efectos de interacciones a bajas energıas provee de los mecanismosdinamicos.

• En todas las teorıas GUT el proton es inestable [13]: se pueden producir decaimientosdel tipo p→ π0e+ en los cuales se viola la conservacion del numero de bariones. Estopermite estudiar la viabilidad de estas teorıas contrastando sus predicciones del tiempode vida medio del proton con el lımite encontrado en el experimento super-Kamiokandeτp→π0e+ > 1033 anos.

La realizacion posible del programa de unificacion en el marco de las teorıas GUT esaparentemente incompleta: no pretenden unificar las interacciones fuertes y electrodebiles conla gravedad, que se puede describir en terminos del campo de la metrica del espaciotiempogµν . La base natural para buscar la unificacion de todas las interacciones fundamentalesW±µ , Zµ, Aµ, A

rµ, gµν esta relacionada con la supersimetrıa [14–17] y la teorıa de cuerdas

(hoy en dıa tambien conocida como teorıa M). Ademas estas podrıan dar cuenta de otrosproblemas. Por ejemplo, algunos candidatos a partıcula de materia oscura son partıculasque surgen de manera natural en teorıas supersimetricas. Mencionamos el neutralino, lapartıcula hipotetica, electricamente neutra, que solo interacciona a traves de la interacciongravitatoria y la debil y que en algunos modelos aparece como mezcla (combinacionlineal) de supercompaneras (”superpartners”) de partıculas del modelo estandar, como el”Higgsino”(del Higgs), los ”Winos” (de W±

µ ) y ”Zinos”(de Zµ).

La gravedad clasica se describe mediante la Teorıa de la Relatividad General. El grupogauge se puede identificar como difeomorfismos, es decir, transformaciones generales de

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coordenadas o en el formalismo de tetradas (”moving frame”) con el grupo de LorentzSO(1, 3). El campo gauge (de espın 2) es la metrica gµν , o las tetradas (”vielbeins”) eaµ, quecomponen la metrica gµν = eaµηabe

bν . La teorıa tambien contiene la conexion de Christoffel

Γρµν(g) o/y un campo gauge para la simetrıa de Lorentz, la conexion de spin ωabµ (e), queson campos compuestos construidos de la metrica o de las tetradas, respectivamente.Resumiendo, la Relatividad General es una teorıa gauge para las simetrıas del espaciotiempo[18].

Como se ha mencionado anteriormente, el modelo estandar tambien es una teorıa decampos gauge, pero con grupo SU(3)× SU(2)× U(1) de simetrias internas. Es decir, esuna teorıa de campos gauge del tipo de Yang-Mills [19].

El teorema de Coleman y Mandula [20] establece que el algebra de Lie mas generalde las simetrıas de la matriz S de una teorıa cuantica de campos contiene el operadorenergıa–momento Pm, los generadores de rotaciones de Lorentz Mmn y un numero finitode operadores Bl que son escalares del grupo de Lorentz. Ademas estos ultimos debenpertenecer al algebra de Lie de un grupo compacto de Lie.Se puede tratar este como un teorema no − go que prohibe unificar las simetrıas delespaciotiempo con las simetrıas internas.

La supersimetrıa evita las restriciones del teorema de Coleman y Mandula [21] general-izando la nocion de algebra de Lie. Sus generadores Qα, Qα son fermionicos y satisfacenunas relaciones de anticonmutacion, y no conmutacion como es el caso de algebras de Liehabituales. Estas nuevas ”algebras” que, involucran tanto conmutadores como anticonmu-tadores, se pueden considerar como construidas con generadores bosonicos y fermionicos, sellaman superalgebras o algebras de Lie graduadas.

Todas las representaciones lineales de la supersimetrıa contienen el mismo numero deestados bosonicos y fermionicos. Esta es la razon de que la busqueda de la supersimetrıa en lanaturaleza se relacione con la busqueda de las supercompaneras de las partıculas elementalesconocidas. En caso de supersimetrıa preservada tienen las mismas caracterısticas que susanalogas ”habituales” pero la estadıstica opuesta: las supercompaneras de los bosones sonfermiones y viceversa. Los datos experimentales sugieren claramente que en la naturalezala supersimetrıa tiene que ser una simetrıa rota, presumiblemente de manera espontanea.La diferencia de masa entre supercompaneras tiene entonces que estar relacionada con laescala de energıa a la que la supersimetrıa se rompe.

Las teorıas supersimetricas tienen unas propiedades muy atractivas. A nivel cuanticoalgunos modelos supersimetricos exhiben un mejor comportamiento ultravioleta. Por ejemplo,ya en el modelo propuesto por Wess y Zumino en 1974 [17] debido a las cancelaciones entrelas contribuciones fermionicas y bosonicas no aparecen divergencias a un lazo (”loop”).Pero sin duda el mejor ejemplo es la teorıa extendida de super–Yang–Mills (SYM) consupersimetrıa extendida N = 4, que nos provee con el unico ejemplo conocido de teorıa decampos 4–dimensionales que es finita a todos los ordenes de la teorıa de perturbaciones[22].

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Como se habıa notado ya en uno de los primeros artıculos de supersimetrıa [15, 16],invariancia bajo supersimetrıa local (”gauge”) implica invariancia bajo difeomorfismos, yentonces, es una teorıa de gravedad. En efecto, el algebra de supersimetrıa sencilla

Qα, Qβ = 0, Qα, Qβ = 0,

Qα, Qα ∼ σaααPa,

implica que dos transformaciones de supersimetrıa producen una traslacion. Entonces, elalgebra de supersimetrıa local se cierra en la traslacion local, es decir en transformacionesde coordenadas locales (invertibles) del tipo mas general, que se llaman difeomorfismos.

En los anos setenta se creıa que la gravedad supersimetrica (supergravedad) [23–28] podıaresolver la tension aparente que existıa entre la teorıa de la Relatividad General de Einsteiny la mecanica cuantica. La esperanza original fue que la supersimetrıa extendida maxima,N = 8, podıa prohibir los contraterminos y hacer que la supergravedad N = 8 [29] notuviera divergencias [30]. Investigaciones mas detalladas mostraron la existencia de posiblescontraterminos en lazos mas altos [31–35]. Curiosamente, una tımida esperanza en lacancelacion de todas las divergencias en supergravedad N = 8 ha reaparecido recientemente[36–43].

Aun ası, la mayorıa de cientıficos creen\esperan mas que la supergravedad cuanticanecesita una completitud ultravioleta, es decir que unos grados de libertad adicionales tienenque manifiestarse en procesos de alta energıa para corregir propiedades de las amplitudes.La teorıa de cuerdas supersimetrica se considera actualmente como dicha completitudultravioleta, siendo su lımite de bajas energıas la supergravedad. Lo que hoy en dıaconocemos como teorıa de supercuerdas o teorıa M es, de un lado, el resultado de incluirsupersimetrıa en la teorıa de cuerdas bosonicas, incorporando ası fermiones y mejorando suespectro de estados cuanticos al eliminar el estado taquionico caracterıstico de la cuerdabosonica.

De otro lado, la teorıa M aparecio como unificacion de los diferentes modelos de super-cuerda. En los anos ochenta del siglo pasado, tras la primera revolucion de supercuerdas, sehabıan desarrollado cinco modelos consistentes de cuerdas supersimetricas:

• Supercuerda tipo I

• Supercuerda tipo IIA

• Supercuerda tipo IIB

• Supercuerda heterotica SO(32)

• Supercuerda heterotica E8 ⊗ E8.

Todas estas teorıas no contienen ningun estado taquionico (a diferencia del modelo decuerdas bosonicas) y ademas son consistentes en 10 dimensiones, es decir, tienen dimension

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crıtica D = 10 en vez de las 26 del modelo bosonico. El hecho de que fueran cinco lasteorıas consistentes se podıa considerar como un problema si lo que se pretende es encontraruna teorıa unica.

Investigaciones mas extensas resultaron en el descubrimiento de las dualidades queconectan los diferentes modelos de cuerdas, y las supergravedades que aparecen en suslımites de bajas energıas, entre ellos y con supergravedad once-dimensional [44].

Ası por ejemplo, las teorıas de tipo IIA y IIB, tras realizar una compactificacion de una delas dimensiones del espaciotiempo en un cırculo, son equivalentes. Las transformacionesque las identifican se conocen como dualidad T. Curiosamente, si el radio de la dimensioncompacta de la teorıa IIA es R, el radio del cırculo en la teorıa equivalente del tipo IIB esα′

R, donde α′ es la pendiente de las trayectorias de Regge J = ~α′M2 + α en las que se

situan los estados cuanticos de la cuerda con espın J y masa M .La otra dualidad, conocida como dualidad S, es 2 una generalizacion de la dualidad que

existe en la electrodinamica clasica entre el campo electrico y el magnetico. A diferenciade la dualidad T, que se ve en teorıa de perturbaciones de la cuerda, la dualidad S esno perturbativa. Se puede observar su manisfestacion como una simetrıa SL(2,R) de lasupergravedad del tipo IIB. En la teorıa cuantica de la cuerda IIB la simetrıa SL(2,R) estarota hasta su subgrupo SL(2,Z) debido a la existencia de otros objetos supersimetricosextendidos, super–p–branas de diferentes tipos, y por la cuantizacion de sus tensiones Tp [46]como consecuencia de una generalizacion de la ”condicion de cuantizacion de Dirac” [47].Este ultimo, en su forma clasica, resulta en la cuantizacion de la carga electrica en el casode la presencia de al menos un monopolo magnetico3. La unidad de todas las dualidades dela teorıa de cuerdas fue apreciada en [44] y resulta en la nocion de dualidad U .

Estos y otros descubrimientos [49–52] sugirieron la conjetura de la teorıa M, una teorıahipotetica que reune los cinco modelos de cuerdas y la supergravedad once–dimensionaly tiene las dualidades que relacionan los modelos de cuerdas como sus simetrıas. Lascinco teorıas de supercuerdas aparecen como distintos lımites perturbativos de esta teorıasubyacente (”underlying”) que ”vive” en 11 dimensiones [53, 54]; y en otro lımite de bajasenergıas se reduce a la supergravedad en 11 dimensiones formulada por E. Cremmer, B.Julia y J. Scherk en 1978 [55].

Tenemos que mencionar que son once las dimensiones ”elegidas” por la teorıa M ya queonce dimensiones es el numero maximo que puede tener una teorıa supersimetrica sin queen su reduccion a 4 dimensiones aparezcan partıculas con espın mayor que 2.

En este aspecto cabe subrayar que la cuerda sı que contiene campos de espın altos ensu espectro de estados cuanticos y que, en relacion con este hecho, existen especulacionessobre que las teorıas de cuerdas pueden ser una fase rota de un modelo mas simetrico decampos conformes de espines altos sin masa [56].

En el estudio del regimen no perturbativo de la teorıa de cuerdas\teorıa M la nocion desuper–p–brana, objeto supersimetrico extendido con dimension de volumen–mundo d = p+1,

2Vease por ejemplo [45]3Vease [47, 48]

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es fundamental. Son estados BPS (Bogomol’nyi-Prasad-Sommerfield), es decir son establesporque saturan un cota BPS (”BPS bound”) que, a su vez, significa que su energıa (odensidad de energıa) tiene un valor minimal entre los estados con valores fijos de algunascargas. La cota BPS, es decir, la restriccion por abajo del valor de la energıa por el valor deuna carga topologica, suele aparecer en la teorıa de cuerdas como resultado de preservar(una parte de) la supersimetrıa. En teorıa de cuerdas\teorıa M los estados de p–branas BPSpueden ser descritos por soluciones supersimetricas de las ecuaciones de supergravedad en10 y 11 dimensiones. La estabilidad mencionada anteriormente, hace que sean similares asoluciones solitonicas de las famosas ecuaciones no lineales de KdV (Korteweg–de Vries) ySine–Gordon [57].

La relacion de la estabilidad de los estados BPS con la topologıa y con la preservacion desupersimetrıa sugieren que las super–p–branas descritas por soluciones supersimetricas delas ecuaciones de supergravedad clasica [48, 58] son objetos de la teorıa completa, es decirde la hipotetica teorıa M cuantica.

Las soluciones supersimetricas de la teorıa de supergravedad en D dimensiones [48, 58]describen los estados fundamentales de las super–p–branas. La dinamica de las excitacionessobre estos estados se describe mediante acciones efectivas de super–p–branas [59–69],similares a la accion de Green y Schwarz para la supercuerda [70]. Los lımites bosonicos deestas acciones se pueden considerar como fuentes de la teorıa de la Relatividad General y delos lımites bosonicos de supergravedad en D dimensiones.

La descripcion dinamica completa de los sistemas de super–p–branas en interaccioncon supergravedad dinamica es un problema mas complicado y no resuelto de una formacompleta4.

En esta tesis estudiamos, entre otros, sistemas en interaccion de supergravedad dinamicay super–p–brana en superespacio simple 4–dimensional. El estudio de este tipo de sis-temas es importante ya que pueden ayudarnos a comprender sistemas mas complicados enespaciotiempo de 10 y 11 dimensiones de la teorıa de cuerdas\teorıa M.

Supergravedad en interaccion con super–p–brana

La accion para la M2–brana en 11 dimensiones se contruyo en [61]. Mas aun, en [61] semostro que la consistencia del acoplo de la supermembrana en un fondo (”background”)de supergravedad (es decir, la existencia de simetrıa–κ en espacio curvo) impone al fondoun conjunto de ligaduras del superespacio que resultan en las ecuaciones del movimientopara los campos fısicos de supergravedad en 11 dimensiones, del mismo modo en que lascondiciones de consistencia para acoplar una supercuerda a supergravedad en supercamposen 10 dimensiones produce las ecuaciones del movimiento para supergravedad D = 10 [75].Es decir, la consistencia del modelo en superespacio curvo de 11 dimensiones requiere quela curvatura y la torsion de este obedezcan las ligaduras de supergravedad que a su vez,resultan en las ecuaciones del movimiento de supergravedad. En este sentido se puede decirque la dinamica de la supergravedad esta gobernada por la M2–brana.

4Vease [71–74] para un progreso parcial en esta direccion.

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Poco despues de los artıculos pioneros [61] se estudio [76] el homologo no trivial mas simplede la M2–brana, la supermembrana D = 4 N = 1. Su autoconsistencia en superespaciocurvo tambien necesita de un conjunto de ligaduras del superespacio. Sin embargo, alcontrario que en el caso de 11 dimensiones estas ligaduras D = 4 N = 1 son off-shellen el sentido de que como consecuencia suya no se producen ecuaciones del movimiento.Esto implica que es posible construir la descripcion Lagrangiana manifiestamente covariantesupersimetrica en supercampos del sistema en interaccion de supergravedad y supermembranaD = 4 N = 1. Curisosamente este sistema no se habıa contruido, por lo que sera parte delestudio de esta tesis5.

Como se ha mencionado anteriormente, los estados fundamentales de las super–p–branasen D dimensiones estan descritos por las soluciones supersimetricas de la teorıa de su-pergravedad en D dimensiones [48, 58]. A pesar de que estas soluciones son puramentebosonicas preservan un medio de la supersimetrıa local caracterıstica de la teorıa de super-gravedad. Esto significa que hay una rotura espontanea parcial de supersimetrıa debido a lapresencia de una super–p–brana [59, 60]. La relacion entre las supersimetrıas preservadas(1/2) por la solucion de la p–brana de supergravedad con la simetrıa–κ de la accion envolumen–mundo para la correspondiente super–p–brana se puso de manifiesto en [77].

En [71] se mostro que el lımite puramente bosonico de la accion de la supermembrana,donde el fermion de Goldstone de la supermembrana (el homologo en el volumen-mundo delGoldstonion de Volkov y Akulov [15, 16]) es puesto a cero, θβ(ξ) = 0, sigue preservandoun medio de la supersimetrıa local de supergravedad. Esta parte preservada (1/2) de lasupersimetrıa del espaciotiempo esta en correspondencia uno a uno con la simetrıa–κ de laaccion completa de la supermembrana [61] y es la simetrıa gauge del sistema en interacciondescrito por la suma de la accion bosonica para la membrana y la accion para supergravedaden 11 dimensiones sin campos auxiliares.

El origen de esta propiedad poco evidente (que no esta restringido unicamente a lasacciones de supergravedad–supermembrana en interaccion en 11 dimensiones, si no que esvalido para una larga coleccion de sistemas dinamicos de super–p–branas y supergravedad)se aclaro en [72, 73] donde se mostro que la suma del lımite puramente bosonico de la accionde la super–p–brana y la accion en componentes espaciotemporales para supergravedad sepuede obtener fijando un gauge en la accion completa en supercampos para el sistema eninteraccion de supergravedad y super–p–brana. Esta ultima accion es la suma de la accioncompleta de la super–p–brana y la accion en supercampos para supergravedad (cuando estaexiste y es conocida).

Es por esto que la descripcion del sistema en interaccion de supergravedad–super–p–branamediante la suma del lımite puramente bosonico de la accion de la p–brana y la accion encomponentes espaciotemporales de supergravedad sin campos auxiliares [71–74] se llamadescripcion ”completa pero con gauge fijo”, donde ”completa” hace referencia al hechode que reproduce todas las ecuaciones del sistema en interaccion, aunque en su versioncon un gauge fijo. Incluso reproduce el lımite θβ(ξ) = 0 de la ecuacion para el fermion de

5Vease capıtulo 3.

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Goldstone de la super–p–brana [72–74]. La descripcion de un sistema dinamico de estetipo para supermembrana D = 4 N = 1 en interaccion con supergravedad y multipletes demateria se desarrollo en terminos de campos ”componentes” en [78].

Ası pues el sistema en interaccion de supergravedad dinamica y super–p–brana puede serestudiado en el marco de la descripcion covariante pero con gauge fijo incluso cuando no seconoce la formulacion en supercampos para supergravedad.

Sin embargo, el estudio en dimensiones mas bajas, en particular en D = 4, de lasecuaciones en supercampos para los sistemas en interaccion de supergravedad y superbranas,cuando es posible, tiene tambien gran interes ya que podrıa proveer nuevos datos sobre laspropiedades de sistemas mas complicados de la teorıa M. En particular, como veremos en elcapıtulo 3, ayuda a comprender el papel de los campos auxiliares de supergravedad en algunossistemas en interaccion de supergravedad y super–p–branas. Ademas estos sistemas son deinteres en si mismos ya que podrıan servir de base para construir modelos fenomenologicosde supergravedad cuadridimensional. En el capıtulo 3 de esta tesis presentamos un estudiocompleto de la descripcion Lagrangiana en supercampos del sistema en interaccion desupergravedad minimal D = 4 N = 1 y supermembrana. Este estudio puede ser consideradocomo el desarrollo de la lınea de investigacion de [72, 73] y [79].

La descripcion Lagrangiana en supercampos para el sistema dinamico de supergravedadD = 4 N = 1 y superpartıcula se desarrollo en [72]. Las ecuaciones en supercampos parael sistema dinamico de supergravedad, supercuerda y supermultiplete tensorial se obtuvieronen [73].

Ambos modelos se han usado para estudiar el origen ası como las propiedades de ladescripcion Lagrangiana completa pero con un gauge fijo del sistema de supergravedady super–p–brana. Esta descripcion propuesta y desarrollada en [71–74] puede ser usadatambien en sistemas de supergravedad mas brana(s) en interaccion en dimensiones masaltas.

Debido a que el sistema de ecuaciones en supercampos para el sistema de supergravedad,supercuerda y supermultiplete tensorial obtenido en [73] resulta ser demasiado complicadopara ser practico, probablemente se podrıa obtener un sistema de ecuaciones menos com-plicado si se usara la formulacion en supercampos de la llamada supergravedad minimal”nueva” [25, 80] en vez de la supergravedad minimal ”vieja” [81, 82] usada en [73]. Estasuposicion esta relacionada con el hecho de que la formulacion minimal ”nueva” incluye untensor antisimetrico auxiliar que tiene un acoplo natural al modelo de cuerda por lo queno es necesario introducir a mano, como en [73], un multiplete tensorial ademas del desupergravedad.

Por otro lado, se pueden usar los resultado de [73] para extraer las ecuaciones ensupercampos para la supercuerda en interaccion con el multiplete tensorial en superespacioplano D = 4 N = 1. La existencia de esa interaccion no trivial esta relacionada con elhecho de que, de acuerdo con [83], se puede usar el multiplete tensorial para construir una3–forma supersimetrica y cerrada en superespacio plano D = 4 N = 1. Es natural empezar

x

estudiando las ecuaciones en supercampos para el sistema formado por la supercuerday el multiplete tensorial antes de pasar al estudio de supergravedad en interaccion consupercuerda pero el sistema de ecuaciones en supercampos en interaccion resulta ser inclusomas sencillo si lo escribimos para un objeto supersimetrico extendido en interaccion con unsupermultiplete escalar.

No es posible hacerlo con la supercuerda pero si con la supermembrana ya que no esposible construir una 3–forma field strength a partir del multiplete escalar, pero una 4–forması. Ası que como primer paso hacia el estudio de la supermembrana en interaccion consupergravedad, estudiaremos en el capıtulo 2 la descripcion Lagrangiana y obtendremos lasecuaciones del movimiento en supercampos para el sistema en interaccion de supermembranaD = 4 N = 1 y multiplete escalar.

El estudio de la descripcion Lagrangiana en supercampos del sistema de supermembranaD = 4 N = 1 y supergravedad comenzo en [84], donde se desarrollo un formalismo deltipo Wess–Zumino para la supergravedad minimal especial de Grisaru–Siegel–Gates–Ovrut–Waldram [85–87] y se encontraron las expresiones para las corrientes (en supercampos)asociadas a la supermembrana que aparecen en el lado derecho de las ecuaciones ensupercampos de supergravedad.

Como veremos en el capıtulo 3 las ecuaciones en supercampos de supergravedad concontribuciones de la membrana son muy complicadas pero se simplifican drasticamente en ungauge especial que llamamos gauge “WZθ=0”. Se llega a este gauge fijando el gauge usualde Wess–Zumino (WZ) para supergravedad y despues usando un medio de las supersimetrıaslocales del volumen–mundo de la supermembrana para fijar el gauge en el cual el fermionde Goldstone de la supermembrana se hace cero, θβ(ξ) = 0.

En este gauge resolvemos las ecuaciones para los campos auxiliares y mostramos que haytres tipos de contribuciones provinientes de la supermembrana que aparecen en las ecuacionesde Einstein del sistema en interaccion. Ademas de los terminos singulares relacionados conel volumen–mundo de la supermembrana W 3, la supermembrana produce dos terminosregulares que pueden ser considerados como contribuciones a la constante cosmologica enlas dos regiones del espaciotiempo separadas por el volumen–mundo de la membrana.

La primera de estas contribuciones, conocida por el estudio [87] (y anteriormente en[88–91], ver [84] para mas referencias y discusion sobre ella), es la constante cosmologicagenerada dinamicamente. La segunda contribucion no singular de la supermembrana, cambiael valor de la constante cosmologica en una de las dos regiones de espaciotiempo haciendoque el valor de las constante cosmologica en las dos ramas del espaciotiempo M4

+ y M4−,

separadas por el volumen mundo de la supermembrana sea, en general, distinto. Esteefecto, que puede ser llamado shift o renormalizacion de la constante cosmologica debidaa las contribuciones de la supermembrana, fue discutido en [88] y [92] en una perspectivapuramente bosonica.

Ası pues genericamente el estado fundamental de nuestro sistema en interaccion describea la supermembrana separando dos espacios de Anti-deSitter con diferentes valores de la

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constante cosmologica. En el contexto puramente bosonico las soluciones de ese tipo fueronestudiadas (ademas de [88] y [92]) en [93–95]. La solucion particular en la que ambasconstantes cosmologicas tienen el mismo valor se puede encontrar en [87]. En esta tesistambien se presenta una discusion de posibles soluciones supersimetricas del sistema eninteraccion con una supermembrana separando dos espacios asintoticamente Anti–deSittercon diferentes valores de la constante cosmologica.

Teorıas de campos de alto espın

Es conocido que las teorıas conformes de campos libres de alto espın en D = 4 pueden serformuladas como una teorıa de campos en un espacio tensorial de diez dimensiones, Σ(10|0),parametrizado por 10 coordenadas bosonicas (xm, ymn) [96–103],

Xαβ = Xβα =1

4xmγαβm +

1

8ymnγαβmn α, β = 1, 2, 3, 4 , m, n = 0, 1, 2, 3 .

y en un superespacio tensorial N = 1, Σ(10|4) con coordenadas (Xαβ, θα) [97–103].

Este espacio tensorial bosonico se propuso como base natural para contruir teorıas decampos conformes de alto espın en D = 4 en [96].

En [104, 105],[106], [96–102] y [103, 107, 108] se han presentado espacios tensoriales

mas generales de dimension n(n+1)2

en el sentido de matrices n× n simetricas Xαβ (α, β =1, . . . , n) que, para n par (n = 4, 8, 16 y 32), determinan la extension del espaciotiempoestandar de dimension D=4,6,10 (D = n

2+ 2) y D = 11.

Anadiendo n coordenadas fermionicas θα se pueden obtener los superespacios tensoriales

”simples” (N = 1) Σ(n(n+1)

2|n),

Σ(n(n+1)

2|n) : ZM := (Xαβ , θα) ,

α, β = 1, ..., n,

Xαβ = Xβα .

que, en su version plana, tambien tienen estructura de supergrupo.

Tomar n par no es una restriccion si uno piensa en el origen espinorial que subyace enlos ındices α; es mas esto motiva la resticcion a n = 2k = 2, 4, 8, 16, ... suponiendo que

θα son espinores. Aunque las coordenadas fermionicas en Σ(n(n+1)

2|n) se suponen reales

normalmente, en el caso n = 4 D = 4 es conveniente considerar θα como un espinor deMajorana en la realizacion de Weyl de las matrices de Dirac por lo que θα =

(θA, θA

).

Para n = 2 las coordenadas espın-tensoriales Xαβ estan expresadas en terminos de lascoordenadas del 3–vector espaciotemporal, Xαβ ∝ xaγαβa ası que Σ(3|2) es simplementeel superespacio usual D = 3 N = 1. El caso n = 32 da como resultado la extensiondel superespacio de 11 dimensiones Σ(528|32), importante en el contexto de la hipotesis de

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los preones BPS [107, 109] y tambien en el analisis de la estructura gauge oculta de lasupergravedad en D = 11 [106, 108].

En discusiones de las teorıas de espines altos, incluso en el capıtulo 4 de esta tesis, seconsideran los casos n = 4, 8, 16 que se usan para describir teorıas conformes de camposde alto espın sin masa en D = 4, 6, 10. Casi todas nuestras ecuaciones en el capıtulo 4seran validas para esas dimensiones, aunque haremos especial enfasis en el caso n = 4 quecorresponde a D = 4.

El primer sistema mecanico en el superespacio tensorial N = 1 D = 4 Σ(10|4) y sus

generalizaciones de dimensiones mas altas Σ(n(n+1)

2|n) con n > 4 se propusieron en [110],

donde se observo que el estado fundamental de ese modelo de superpartıcula describe unestado BPS que preserva todas las supersimetrıas excepto una. El posible papel como”contituyentes” de estos estados en la teorıa de cuerdas/M se introdujo y se discutio engeneral en [109], donde se les llamo ”preones BPS” ( vease tambien [107]). Ası que desdeese punto de vista, la superpartıcula en [110] podrıa ser llamada ”preonica”. Su cuantizacionse desarrollo en [97], donde se mostro que el espectro de la superpartıcula preonica n = 4cuantizada se puede describir con una torre de campos conformes de alto espın sin masa detodas las helicidades posibles y se presentaron evidencias de que los modelos con n = 8 yn = 16 describen teorıas conformes de alto espın en espaciotiempos de 6 y 10 dimensiones.

En [98, 99] se presento y estudio una forma elegante de las ecuaciones de alto espınbosonicas y fermionicas en el espacio tensorial Σ(10|0). Mientras que en [103] se dio la formaexplıcita de las ecuaciones conformes de alto espın en espacios tensoriales con D = 6, 10.Las ecuaciones en supercampos para los supermultipletes de campos conformes sin masa dedimension D = 6, 10 en los superespacios tensoriales N = 1 Σ(36|8) y Σ(136|16),

D = 10 Σ(136|16) : ZM := (Xαβ , θα) ,

α, β = 1, ..., 16,

Xαβ = 116xmσαβm + 1

2·5!ym1...m5σαβm1...m5

,

m = 0, 1, . . . 9

fueron propuestas en [102].

En particular, los supermultipletes N = 1 de campos conformes de alto espın en D =4, 6, 10 estan descritos por supercampos escalares en los correspondiente superespacios

n=4,8,16 Σ(n(n+1)

2|n),

Φ(Xαβ, θγ) = b(X) + fα(X) θα +n∑i=2

φα1···αi(X) θα1 · · · θαi ,

obedeciendo la ecuacion [102]

D[αDβ]Φ(X, θ) = 0 .

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Aquı,

Dα =∂

∂θα+ iθβ∂βα , Dαβ = ∂αβ :=

∂

∂Xαβ,

son derivadas covariantes en el superespacio tensorial rıgido Σ(n(n+1)

2|n) que satisfacen

Dα, Dβ = 2i∂αβ

donde se puede ver que exhiben la estructura de extension central de las superalgebras delos superespacios tensoriales [111] (vease [112]).

Las ecuaciones de alto espın en superespacio tensorial con supersimetrıa N -extendida,han sido estudiadas en [113, 114]6 que constituyen la base del capıtulo 4 de esta tesis.

En el capıtulo 4 presentamos las ecuaciones supersimetricas conformes de alto espın libres

con N = 2, N = 4 y N = 8 en superespacios tensoriales N -extendidos Σ(n(n+1)

2|N n). Entre

otros, vamos a mostrar que las ecuaciones conformes de alto espın de los supermultipletesN = 2 de D = 4, 6, 10 estan descritas por supercampos escalares quirales Φ(Xαβ,Θγ, Θγ)

en el superespacio tensorial N = 2 Σ(n(n+1)

2|2n), que obedecen el siguiente conjunto de

ecuaciones lineales en supercampos,

DαΦ = 0 , D[αDβ]Φ = 0 ,

donde

Dα =∂

∂Θα+ iΘβ∂βα =

1

2(Dα1 + iDα2) ,

Dα =∂

∂Θα+ iΘβ∂βα = −(Dα)∗ , Dαβ = ∂αβ :=

∂

∂Xαβ,

son las derivadas covariantes en el superespacio rıgido N = 2–extendido Σ(n(n+1)

2|2n), que

obedecen el superalgebra

Dα,Dβ = 0 , Dα, Dβ = 2i∂αβ , Dα, Dβ = 0 .

Presentamos tambien las ecuaciones en supercampos para superespacios tensoriales extendi-dos con N par y mayor que 2 que generalizan las ecuaciones de N = 2.

Tambien presentaremos la generalizacion supersimetrica extendida N > 1 del modelo desuperpartıcula preonica de [97] y mostraremos como se pueden obtener las ecuaciones de

los supercampos de alto espın cuantizando el modelo de superpartıcula en Σ(n(n+1)

2|N n) para

N ≥ 2 con N par.

6Vease [115, 116] para una descripcion de las ecuaciones supersimetricas de alto espın libres en superespaciousual y [117, 118] para la version en supercampos de las ecuaciones de campos de alto espın de Vasilieven interaccion [119–121].

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Aunque nuestras ecuaciones son validas para cualesquiera N y n par, elaboraremosen detalle los casos N = 2, 4, 8, que, para n = 4, corresponden a los supermultipletessin masas de campos de alto espın en D = 4, los cuales estan en correspondencia claracon las teorıas estandar de campos de ”bajo espın”. Estas son el hipermultiplete paraN = 2, el supermultiplete supersimetrico de Yang–Mills para N = 4 y el multiplete desupergravedad maximal para N = 8, el cual en su version linearizada se puede describirmediante supercampos escalares en superespacios estandar extendidos en D = 4 Σ(4|4N )

con N = 2, 4, 8.

Una de las razones de nuestro interes en sistemas N -extendidos supersimetricos de teorıasde alto espın venıa de la observacion de que la supersimetrıa N -extendida con N = 4 unificalos campos gauge escalares y vectoriales. Por otro lado, todas las ecuaciones de alto espınhan sido formuladas en terminos de campos escalares y espinoriales en espacio tensorial asıque el estudio de supersimetrıas N -extendidas podrıa ser interesante para buscar posiblesgeneralizaciones de las ecuaciones de Maxwell y Einstein en superespacios tensoriales.

De hecho, en cierto punto de nuestro estudio de las ecuaciones en supercampos ensuperespacio tensorial N = 4 Σ(10|16) aparecen los analogos de las ecuaciones de Maxwell enespacio tensorial . Sin embargo, un analisis mas detallado muestra que estos campos (espın)–tensoriales se pueden expresar como derivadas de otros campos escalares en superespaciotensorial ası que, por ejemplo, el sector bosonico del multiplete de alto espın conformeN = 4 esta expresado por dos campos escalares complejos en el espacio tensorial, φ y φ.

Del mismo modo, cuando estudiamos las ecuaciones en supercampos en superespaciotensorial N = 8, Σ(10|32), a pesar de que en cierta etapa aparecen las generalizacionesen espacio tensorial de las ecuaciones de (super)gravedad conforme, se demuestra quese reducen a las ecuaciones para campos escalares (y espinoriales) en espacio tensorialpresentadas por primera vez por Vasiliev [98]. En cierto sentido se puede decir que elresultado de aumentar N , es que aparecen mas campos escalares y espinoriales.

Sin embargo, para N = 4 aparece un nuevo fenomeno. Como el nuevo campo escalaraparece en la teorıa unicamente a traves de campos de tipo de Maxwell, es decir bajoderivadas bosonicas ∂αβ, la teorıa se hace invariante bajo desplazamientos constantes de esecampo bosonico que hace que el segundo campo escalar sea similar a los axiones (para loscuales dicha simetrıa se llama simetrıa de Peccei–Queen [122]7). En el multiplete N = 8reformulado en terminos de campos, la simetrıa de Peccei–Queen se hace mas complicadapara los campos escalares e incluso esta presente en los campos espinoriales que entranen el modelo bajo la accion de una derivada simulando la estructura de los campos deRarita-Schwinger.

7Dado que en la teorıa de cuerdas de tipo IIB y en supergravedad IIB el axion aparece como un miembrode la familia de los campos gauge RR, su simetrıa de Peccei–Queen puede ser considerada como elhomologo de la simetrıa gauge caracterıstica de los potenciales gauge RR altos.

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Sistema de multiples M0–branas

En [50] se motivo que una descripcion aproximada de un sistema de p–branas de Dirichlet(Dp–branas) quasi coincidentes viene dada por una teorıa de Yang–Mills maximal super-simetrica (SYM) con grupo de gauge U(N), la cual puede ser obtenida mediante reducciondimensional de la teorıa de SYM en D=10 con U(N) a d = p+ 1. Esta incluye D − p− 1matrices Hermıticas con campos escalares cuyos elementos diagonales describen las diferentesposiciones de las Dp-branas mientras que los elementos no diagonales dan cuenta de lascuerdas que unen las diferentes Dp-branas.

Como es sabido que una sola Dp-brana esta descrita por la suma de la accion supersimetricade Dirac–Born–Infeld, proporcionando una generalizacion no lineal de la accion de Yang–Millscon U(1), y un termino de Wess-Zumino (vease [63–67, 123, 124]), era natural buscar unageneralizacion no lineal de la accion de SYM no–abeliana que proporcione una descripcion nolineal mas completa del sistema de Dp-branas quasi coincidentes. Para el lımite bosonico demultiples Dp-branas quasi coincidentes (sistema mDp) la descripcion mas popular esta dadapor la accion de ”branas dielectricas” de Myers [125]. Esta se obtuvo a traves de una seriede transformaciones de dualidad T a partir de la 10D accion no–abeliana de Born–Infeldcon traza simetrica, propuesta por Tseytlin [126] para el lımite puramente bosonico delsistema de multiples D9–branas (sistema mD9) que llenan todo el espaciotiempo. Ambasacciones [125] y [126] han resistido todos los intentos de construir sus generalizacionessupersimetricas durante muchos anos. Ademas, la accion de Myers no tiene simetrıa deLorentz.

La descripcion supersimetrica y covariante Lorentz del sistema de mDp se obtuvo en [127]en el marco del llamado ”enfoque de fermiones de frontera”. Sin embargo, esta descripcionviene dada en lo que se llama ”cuantizacion en nivel menos uno” que significa que parallegar a una descripcion del sistema de mDp similar al del de Dp–branas (como por ejemploel de [65]), se tiene que realizar una cuantizacion del sistema dinamico. Esta tarea no estrivial y no se ha resuelto de manera completa aun, lo que ha motivado un gran numero deintentos de obtener una descripcion aproximada pero covariante bajo el grupo de Lorentz ysupersimetrica del sistema de mDp que vaya mas alla de la aproximacion de SYM (veasepor ejemplo [128]). Solo para el caso del sistema de mD0 existe un candidato no lineal,supersimetrico e invariante Lorentz en D = 10 para la accion de mD0 [129, 130].

Debido a que las Dp–branas con p = 0, 2, 4 pueden ser obtenidas mendiante una reducciondimensional de las branas M0, M2 y M5 en D = 11, es logico suponer que se pueda obtenerel sistema de mDp de su respectivo sistema de mMp.

Sin embargo, para el caso del sistema de mM5 incluso la cuestion de cual es el analogode la descripcion aproximada de muy bajas energıas de SYM aun no se conoce a cienciacierta (vease por ejemplo [131–133] para estudios relacionados y referencias). Para el casodel sistema de mM2 branas dicho problema ha permanecido sin solucion muchos anospero recientemente el d = 3 N = 8 modelo de Bagger, Lambert y Gustavsson (BLG)supersimetrico [134] basado en 3–algebras (vease [135] y referencias allı) en vez de algebras

xvi

de Lie y un modelo de Aharony, Bergman, Jafferis y Maldacena (ABJM) mas convencional[136] (con simetrıa de gauge SU(N)× SU(N) y solo N = 6 supersimetrıas manifiestas)han sido propuestos para dicho papel.

Si se trata del sistema de multiples M0 branas, se construyo un candidato puramentebosonico en [137, 138] generalizando la accion de Myers de la D0–brana dielectrica de 11dimensiones. Por otro lado, se obtuvieron las ecuaciones del movimiento aproximadas perosupersimetricas e invariantes bajo el grupo de Lorentz para el sistema de mM0 en [139] enel marco del enfoque de superembedding (vease [140, 141] ası como [142, 143] y referenciasallı).

La generalizacion de esas ecuaciones para el sistema de mM0 en superespacio curvo desupergravedad 11 dimensional que describe la generalizacion de la ”teorıa de M(atrices)”[144] (vease [145]) para el caso de su interaccion con un fondo de supergravedad arbitrario,se presento y estudio en [146]. En [147] se mostro que en el caso de que el fondo fuerade ondas pp (pp–wave), esas ecuaciones reproducen (en cierta aproximacion) el llamadomodelo BMN de matrices propuesto para ese fondo por Berenstein, Maldacena y Nastaseen [148].

Este resultado confirma que las ecuaciones de [146] describen la teorıa de Matrices inter-accionando con un fondo de supergravedad. Sin embargo, debido al origen de superespaciode dichas ecuaciones, sus apliaciones incluso en un fondo de supergravedad puramentebosonico son extremadamente complicadas. Para este fin se necesita por lo primero encontrarla solucion completa en supercampos de las ligaduras de supergravedad 11D [149] querepresentan la solucion bosonica supersimetrica de las ecuaciones de supergravedad en elespaciotiempo. Esto hizo que fuera deseable encontrar una accion que reprodujera lasecuaciones del modelo de Matrices de [146] o sus generalizaciones.

Para el caso del sistema de mM0 en superespacio plano esta accion se propuso en [150],donde se mostro que posee supersimetrıa local N = 16 1d. En el capıtulo 5 se derivaran yestudiaran las ecuaciones del movimiento del sistema de mM0 descrito por dicha accion.Se estudiaran las soluciones supersimetricas de dichas ecuaciones mostrando que su sectorrelacionado con el centro de energıa es similar a la solucion de las ecuaciones para unaunica M0 brana y se presentaran tambien dos ejemplos de soluciones no supersimetricas condiferentes propiedades del movimiento del centro de energıa.

Contenido de la tesis

El primer capıtulo contiene una introduccion al superespacio ası como a los supercamposy superformas, ya que a lo largo de la tesis se hara uso de dichos conceptos. Tras estabreve exposicion se explicara la geometrıa de los superespacios plano y curvo, para pasar adescribir la supergravedad. Se da una descripcion de la supergravedad minimal, su accion ensuperespacio, ası de como obtener sus ecuaciones del movimiento a partir de esta accion ensupercampos. Aunque a priori no es trivial encontrar las variaciones de los supervielbeins ,

xvii

ya que estos estan restringidos por las ligaduras de supergravedad, es posible encontrarvariaciones admisibles que preservan dichas ligaduras [72, 82]. Terminamos el capıtulo 1presentando estas variaciones admisibles para la supergravedad minimal, que usaremos en elcapıtulo 3.

En el capıtulo 2 presentamos la accion en supercampos para el sistema dinamico desupermembrana D = 4 N = 1 en interaccion con multiplete escalar y la usamos paraobtener las ecuaciones del movimiento en supercampos de este sistema.

Estas incluyen las ecuaciones de la supermembrana que coinciden formalmente conlas ecuaciones de esta en un fondo de campo escalar (off-shell) y las ecuaciones para elsupercampo quiral especial con fuente producida por la supermembrana.

En el caso en el cual la parte de la accion correspondiente al supermultiplete escalar contieneunicamente el termino cinetico mas simple, extraemos las ecuaciones en componentes apartir de las ecuaciones en supercampos y las resolvemos a orden principal en la tension dela supermembrana.

Tambien se discute la inclusion de un superpotencial no trivial y su relacion con lassoluciones supersimetricas del tipo domain wall conocidas.

El capıtulo 3 esta dedicado a obtener el conjunto completo de ecuaciones del movimientopara el sistema en interaccion de supermembrana y supergravedad dinamica D = 4 N =1 variando la accion completa en supercampos. Una vez obtenidas, escribimos estasecuaciones en supercampos en el gauge especial “WZθ=0” donde el campo de Goldstone

de la supermembrana se ha puesto a cero (θ = 0) y ademas esta fijado el gauge deWess–Zumino en los supercampos de supergravedad.

En este gauge resolvemos las ecuaciones para los campos auxiliares y discutimos el efectode la generacion dinamica de la constante cosmologica en la ecuacion de Einstein del sistemaen interaccion y su renormalizacion debida a contribuciones regulares de la supermembrana.Estos dos efectos (descritos por primera vez en los anos 70 y 80 en un contexto bosonico yen la literatura de supergravedad) resultan en que la solucion describe en el caso generico,dos espaciotiempos con distintas constantes cosmologicas separados por el volumen–mundode la supermembrana.

Continuamos en el capıtulo 4 proponiendo las ecuaciones en supercampos en superespaciostensoriales N -extendidos para describir las generalizaciones supersimetricas con N = 2, 4, 8de las teorıas conformes libres de alto espın. Describimos tambien como se puede obtenerestas ecuaciones cuantizando un modelo de superpartıcula en superespacios tensorialesN -extendidos.

Mostramos que los supermultipletes de alto espın N -extendidos contienen unicamentecampos escalares y espinoriales en el espacio tensorial, ası que a diferencia del enfoqueestandar de supercampos en superespacios habituales, no aparecen generalizaciones notriviales de las ecuaciones de Maxwell y Einstein cuando N > 2. Para N = 4, 8 lascomponentes de tipo espın-tensor mas altas del supercampo tensorial extendido se expresana traves de campos escalares y espinoriales adicionales que obedecen las mismas ecuacioneslibres de alto espın, pero estos son del tipo axion en el sentido de que poseen simetrıas detipo Peccei–Quinn.

xviii

En el ultimo capıtulo 5 estudiamos las propiedades de la accion covariante supersimetricay con simetrıa–κ de N M0–branas quasi coincidentes (sistema mM0) en superespacio planode once dimensiones y obtenemos las ecuaciones supersimetricas para este sistema dinamico.

A pesar de que una unica M0–brana se corresponde con la superpartıcula sin masa en 11dimensiones, el movimiento del centro de energıa del sistema de mM0 esta caracterizadopor una masa M no negativa. Esta masa esta construida a partir de los campos matricialesque describen el movimiento relativo de los constituyentes del sistema de mM0.

Mostramos que cualquier solucion bosonica del sistema de mM0 puede ser supersimetricasi y solo si esta masa efectiva se anula, M2 = 0, y que todas las soluciones bosonicassupersimetricas preservan un medio de las supersimetrıas de once dimensiones. Presentamostambien unas soluciones no supersimetricas con M2 6= 0 y discutimos unas propiedadespeculiares del sistema de mM0.

xix

LIST OF PUBLICATIONS

Superfield equations for the interacting system of D=4 N=1 supermembrane andscalar multipletIgor A. Bandos, Carlos Meliveo, Nucl. Phys. B849, 1-27 (2011).

Extended supersymmetry in massless conformal higher spin theoryIgor A. Bandos, Jose A. de Azcarraga, Carlos Meliveo, Nucl.Phys. B853, 760-776 (2011).

Three form potential in (special) minimal supergravity superspace and superme-mbrane supercurrentIgor A. Bandos, Carlos Meliveo, J. Phys. Conf. Ser. 343 012012 (2012).

Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motionIgor A. Bandos, Carlos Meliveo, JHEP 1208 (2012) 140.

Conformal higher spin theory in extended tensorial superspaceIgor A. Bandos, Jose A. de Azcarraga, Carlos Meliveo, Fortsch.Phys. 60, 861-867 (2012).

On supermembrane supercurrent and special minimal supergravityIgor A. Bandos, Carlos Meliveo, Fortsch.Phys. 60, 868-874 (2012).

Covariant action and equations of motion for the eleven dimensional multipleM0-brane systemIgor A. Bandos, Carlos Meliveo, Phys.Rev. D 87 126011 (2013).

xxi

CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

1. Introduction 1

1.1. Flat D = 4 N = 1 superspace . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Superfields on D = 4 N = 1 superspace . . . . . . . . . . . . . . . . . . 2

1.3. Differential Superforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4. Vielbein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.1. Flat Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5. Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1. Minimal Supergravity . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.2. Superfield supergravity action and admissible variation of constrainedsupervielbein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield 11

2.1. Free supermembrane in flat D = 4 N = 1 superspace . . . . . . . . . . . 12

2.2. Scalar supermultiplet as described by chiral superfield . . . . . . . . . . . 14

2.2.1. Four form field strength constructed from the scalar supermultiplet 15

2.3. Supermembrane action in the scalar multiplet background . . . . . . . . . 16

2.3.1. Equations of motion for supermembrane in a background of anoff–shell scalar supermultiplet . . . . . . . . . . . . . . . . . . . . 17

2.4. Superfield equations for the dynamical system of special scalar supermultipletinteracting with supermembrane . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1. Special scalar multiplet and its dual three form potential . . . . . . 18

2.4.2. Superfield equations of motion for interacting system . . . . . . . . 21

2.4.3. Supermembrane current . . . . . . . . . . . . . . . . . . . . . . . 23

xxiii

Contents

2.5. Simplest equations of motion for spacetime fields interacting with dynamicalsupermembrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.1. General structure of the simplest special scalar multiplet equations

with a superfield source . . . . . . . . . . . . . . . . . . . . . . . 242.5.2. Dynamical scalar multiplet equations with supermembrane source

contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.3. Simplest solution of the dynamical equations at leading order in

supermembrane tension . . . . . . . . . . . . . . . . . . . . . . . 272.5.4. Superfield equations with nontrivial superpotential . . . . . . . . . 28

3. Supermembrane interaction with dynamical D=4 N=1 supergravity. Su-perfield Lagrangian description and spacetime equations of motion 313.1. Superfield supergravity and supermembrane in curved D=4, N=1 superspace 32

3.1.1. Superfield supergravity action, superspace constraints and equationsof motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2. Supermembrane action in minimal supergravity background . . . . 333.1.3. 3–form potential in the minimal supergravity superspace. Special

minimal supergravity . . . . . . . . . . . . . . . . . . . . . . . . 363.2. Superspace action and superfield equations of motion for the interacting

system of dynamical supergravity and supermembrane . . . . . . . . . . . 383.3. Spacetime component equations of the D = 4 N = 1 supergravity–

supermembrane interacting system . . . . . . . . . . . . . . . . . . . . . 393.3.1. Wess–Zumino gauge plus partial gauge fixing of the local spacetime

supersymmetry (WZθ=0 gauge) . . . . . . . . . . . . . . . . . . . 393.3.2. Current superfields in the WZθ=0 gauge. Current prepotentials and

Rarita–Schwinger equation . . . . . . . . . . . . . . . . . . . . . . 413.3.3. Einstein equation of the supergravity–supermembrane interacting

system in the WZ θ=0 gauge . . . . . . . . . . . . . . . . . . . . . 423.3.4. Cosmological constant generation in the interacting system and its

“renormalization” due to supermembrane . . . . . . . . . . . . . . 433.4. On supersymmetric solutions of the interacting system equations . . . . . 45

4. Conformal higher spin theory 494.1. Superparticle in N -extended tensorial superspace . . . . . . . . . . . . . . 49

4.1.1. An action for the Σ(n(n+1)

2|N n) superparticle . . . . . . . . . . . . . 49

4.1.2. Symplectic supertwistor form of the action . . . . . . . . . . . . . 504.1.3. Gauge symmetries of the original action . . . . . . . . . . . . . . . 514.1.4. Constraints and their conversion to first class . . . . . . . . . . . . 52

4.2. Quantization of the superparticle in Σ(n(n+1)

2|Nn) with even N and conformal

higher spin equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3. From the superfield to the component form of the higher spin equations in

tensorial space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.1. N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.2. N = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xxiv

Contents

4.3.3. N = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5. Covariant action and equations of motion for the 11D system of multipleM0-branes 635.1. Single M0–brane in spinor moving frame formulation . . . . . . . . . . . . 64

5.1.1. Twistor–like spinor moving frame action and its irreducible κ–symmetry 64

5.1.2. Irreducible κ–symmetry of the spinor moving frame action . . . . . 65

5.1.3. Moving frame and spinor moving frame . . . . . . . . . . . . . . . 66

5.1.4. Cartan forms, differentiation and variation of the (spinor) movingframe variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.5. K9 gauge symmetry of the spinor moving frame action of the M0–brane 69

5.1.6. Derivatives and variations of the Cartan forms . . . . . . . . . . . 70

5.2. Equations of motion of a single M0–brane and induced N = 16 supergravityon the worldline W 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1. All supersymmetric solutions of the M0 equations describe 1/2 BPSstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3. Covariant action for multiple M0–brane system . . . . . . . . . . . . . . . 74

5.3.1. Variables describing the mM0 system . . . . . . . . . . . . . . . . 74

5.3.2. First order form of the 1d N = 16 SYM Lagrangian as a startingpoint to build mM0 action . . . . . . . . . . . . . . . . . . . . . . 75

5.3.3. Induced supergravity on the center of energy worldline . . . . . . . 76

5.3.4. A way towards mM0 action . . . . . . . . . . . . . . . . . . . . . 77

5.3.5. mM0 action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.6. Local supersymmetry of the mM0 action . . . . . . . . . . . . . . 79

5.4. mM0 equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1. Equations of the relative motion . . . . . . . . . . . . . . . . . . . 80

5.4.2. A convenient gauge fixing . . . . . . . . . . . . . . . . . . . . . . 81

5.4.3. By pass technical comment on derivation of the equations for thecenter of energy coordinate functions . . . . . . . . . . . . . . . . 82

5.4.4. Equations for the center of energy coordinate functions . . . . . . 83

5.4.5. Noether identities for gauge symmetries. First look. . . . . . . . . 84

5.4.6. Equations which follow from the auxiliary field variations and simpli-fication of the above equations . . . . . . . . . . . . . . . . . . . 84

5.4.7. Noether identity, remnant of the K9 gauge symmetry and the finalform of the Ω#i equation . . . . . . . . . . . . . . . . . . . . . . 86

5.5. Ground state solution of the relative motion equations . . . . . . . . . . . 87

5.5.1. Ground state of the relative motion . . . . . . . . . . . . . . . . . 88

5.5.2. Solutions with M2 = 0 have relative motion in the ground state sector 88

5.6. Supersymmetric solutions of mM0 equations . . . . . . . . . . . . . . . . 89

5.6.1. Supersymmetric solutions of the mM0 equations have M2 = 0 . . . 89

5.6.2. All BPS states of mM0 system are 1/2 BPS . . . . . . . . . . . . 90

5.7. On solutions of mM0 equations with M2 > 0 . . . . . . . . . . . . . . . 90

5.7.1. Purely bosonic equations in the case of M2 > 0 . . . . . . . . . . 91

xxv

Contents

5.7.2. Remnant of K9 symmetry in the bosonic limit of the mM0 actionand Ω#i equations . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.7.3. Center of energy velocity and momentum for M2 6= 0 . . . . . . . 935.7.4. An example of non-supersymmetric solutions . . . . . . . . . . . . 945.7.5. Another non-supersymmetric formal solution . . . . . . . . . . . . 96

6. Conclusions 99

A. Notation, conventions and some useful formulae in D = 4 103

B. Supermembrane current superfield which enters the scalar multiplet equa-tions 105

C. On admissible variations of superfield supergravity 111

D. Supermembrane current superfields entering the superfield SUGRA equa-tions 113

E. Useful formulae involving 11-dimensional and 9–dimensional Gamma Ma-trices 117

F. Moving frame and spinor moving frame variables 119

G. Equations of motion for a single M0–brane 123

H. Multiple M0 equations of motion 125

Bibliography 127

xxvi

CHAPTER 1

INTRODUCTION

In field theory supersymmetry transforms bosonic fields into fermionic fields and viceversa. Hence, the irreducible representations of supersymmetry, supermultiplets, involvesome number of bosonic and fermionic fields. A compact and elegant way to describe

supermultiplets is provided by superfield approach.Superspace is an extension of the ordinary spacetime including, besides the usual spacetime

coordinates xµ, extra anticommutative (or fermionic) coordinates. In the case of flat N–extended D = 4 superspace these fermionic coordinates are collected in N two–componentWeyl spinors θαi = (θαi )∗.

Chapters 2, 3 are devoted to the study of the D = 4 N = 1 supermembrane, which is amembrane moving in simple (N = 1) D = 4 superspace, so let us begin reviewing someproperties of this superspace.

1.1. Flat D = 4 N = 1 superspace

We denote the coordinates of flat D = 4 N = 1 superspace Σ(4|4) by

zM = (xµ, θα, θα) (1.1)

where µ = 0, 1, 2, 3 is a 4–vector index, α = 1, 2 and α = 1, 2 are spinorial indices; the”supervector” index M represents both types of indices, M = (µ, α, α).

The spinorial coordinates anticommute, θαθα = −θαθα, θαθβ = −θβθα, θαθβ = −θβ θα,while the bosonic coordinates commute among themselves, xµxν = xνxµ, and also withfermionic coordinates xµθα = θαxµ. These properties can be collected in

zMzN = (−)ε(N)ε(M)zNzM (1.2)

where ε(M) ≡ ε(zM) is the so–called Grassmann parity, ε(µ) = 0, ε(α) = 1 = ε(α).

1

1.2. Superfields on D = 4 N = 1 superspace

1.2. Superfields on D = 4 N = 1 superspace

Superfields are functions defined on superspace which means that superfields depend on bothbosonic and fermionic coordinates. As far as fermionic coordinates anticommute amongthemselves, they are nilpotent, θ′θ′ = −θ′θ′ = 0. This property implies that the seriesexpansion of superfield in fermionic coordinates contains a finite number of terms. Thecoefficients in that series, called superfield components, are the ordinary spacetime fields

F (z) = F (x, θ, θ) = f(x) + θφ(x) + θχ(x) + θθm(x) + θθn(x)

+θσmθvm(x) + θθθλ(x) + θθθψ(x) + θθθθd(x). (1.3)

The statistics (Grassmann parity) of the component fields appearing as a coefficient for evenand odd powers of θ, θ are different (bosonic versus fermionic or vice versa) and dependson the statistics (Grassmann parity) of the superfield.

The rigid D = 4 N = 1 supersymmetry is realized in superspace as constant translationsof the fermionic coordinate, δθ = ε, δθ = ε, supplemented with the following transformationsof bosonic coordinates, δxµ = −iεσµθ + iθσµε (see [15, 16]), to resume,

δxµ = −iεσµθ + iθσµε

δθα = εα

δθα = εα. (1.4)

In general a superfield is a highly reducible representation of the supersymmetry. To extractan irreducible representation one usually needs to impose on superfield some equations interms of fermionic covariant derivatives (see bellow) called constraints. One distinguishes on–shell and off–shell constraints. The on–shell constraints restrict the field content of superfieldto physical fields and impose on these equations of motion. The off–shell constraints do notimpose on physical fields equations of motion and also leave nonvanishing not only physicalcomponent fields of the superfield, but also so–called auxiliary fields, the presence of whichprovides off-shell closure of the (super)algebra of supersymmetry transformations.

1.3. Differential Superforms

The convenience of differential forms is that they are manisfestly invariant under coor-dinate transformations. Let us introduce differentials of superspace coordinates, dzM =(dxµ, dθα, dθα) and the exterior product of these

dzM ∧ dzN = −(−)(ε(N)+1)(ε(M)+1)dzN ∧ dzM = (−)ε(N)ε(M)+1dzN ∧ dzM

dzMzN = −(−)ε(N)ε(M)zNdzM , (1.5)

with ε(N) defined in (1.2). This implies

dxµ ∧ dxν = −dxν ∧ dxµ, dθα ∧ dxµ = −dxµ ∧ dθα,dθα ∧ dθβ = +dθβ ∧ dθα, etc. (1.6)

2

Chapter 1. Introduction

The differential p–form in superspace has then the following structure

Ωp =1

p!dzM1 ∧ .... ∧ dzMpΩMp...M1(z) (1.7)

where ΩMp...M1(z) is a superfield carrying supervector indices. Being contracted with exteriorproduct of dzM , ΩMp...M1(z) has to be graded–antisymmetric, ΩMp...M1](z), which implies

Ω...µ...ν...(z) = −Ω...ν...µ...(z),

Ω...µ...β...(z) = −Ω...β...µ...(z),

Ω...α...β...(z) = +Ω...β...α...(z). (1.8)

If the form Ωp is bosonic, the superfields ΩMp...M1(z) with odd number of spinorial indiceswill have a fermionic behavior while those with even number will have a bosonic one.

With these definitions the exterior product of differential superforms obey

(c1Ωp + c2Ωp′) ∧ Ωq = c1Ωp ∧ Ωq + c2Ωp

′ ∧ Ωq

Ωp ∧ Ωq = (−)pq+ε(Ωp)ε(Ωq)Ωq ∧ Ωp

Ωp ∧ (Ωq ∧ Ωr) = (Ωp ∧ Ωq) ∧ Ωr. (1.9)

Once defined the superforms we introduce the exterior derivative, an operator which mapsp-forms into (p+1)-forms

dΩp =1

p!dzM1 ∧ .... ∧ dzMp ∧ dzN ∂

∂zNΩMp...M1(z) =

=1

(p+ 1)!dzM1 ∧ .... ∧ dzMp+1(∂Mp+1ΩMp...M1(z) + (−)ε(N)ε(M)+1cyclic permutations).

(1.10)

Some useful properties involving exterior derivative one

dd = 0,

d(c1Ωp + c2Σp) = c1dΩp + c2dΣp,

d(Ωp ∧ Σq) = Ωp ∧ dΣq + (−)qdΩp ∧ Σq. (1.11)

1.4. Vielbein

Supervielbeins are 1-forms which define a supersymmetric generalization of local referenceframe. We denote the bosonic and fermionic supervielbein one forms of Σ(4|4) by

Ea = dZMEaM(Z) , Eα = dZMEα

M(Z) , Eα = dZM EMα(Z) , (1.12)

a = 0, 1, 2, 3 , α = 1, 2 , α = 1, 2 .

3

1.4. Vielbein

Sometimes it is convenient to collect them in

EA = (Ea, Eα) = (Ea, Eα, Eα) = dZMEAM(Z) , (1.13)

where α = 1, 2, 3, 4 can be understood as Majorana spinor index.

In superspace the torsion 2–forms are defined as the covariant exterior derivatives of thebosonic and fermionic supervielbein forms

T a := DEa = dEa − Eb ∧ wba =1

2EB ∧ ECTCB

a (1.14)

Tα := DEα = dEα − Eβ ∧ wβα =1

2EB ∧ ECTCB

α , wβα :=

1

4wabσabβ

α , (1.15)

T α := DEα = dEα − Eβ ∧ wβα =

1

2EB ∧ ECTCB

α , wβα :=

1

4wabσab

αβ , (1.16)

where wab = −wba = dZMwabM(Z) is the spin connection 1-form , σabβα = σ[aσb] and

σabαβ = σ[aσb] are antisymmetrized products of the relativistic Pauli matrices (see AppendixA) and d, ∧ are exterior derivative and exterior product of differential forms previously definedin section 1.3. This later is antisymmetric for bosonic one forms, Ea ∧ Eb = −Eb ∧ Ea,symmetric for two fermionic one forms, Eα ∧ Eβ = Eβ ∧ Eα, and again antisymmetric forthe product of bosonic and fermionic one forms, Ea ∧ Eβ = −Eβ ∧ Ea (see 1.9).

The torsion 2-forms obey the Bianchi identities

DT a + Eb ∧Rba = 0 , DTα + Eβ ∧Rβ

α = 0 , DT α + Eβ ∧Rβα = 0 , (1.17)

where

Rab = (dw − w ∧ w)ab =1

2EB ∧ ECRCB

ab (1.18)

is the curvature 2-form. Its Bianchi identities read DRab = 0.

1.4.1. Flat Superspace

In flat D = 4 N = 1 Σ(4|4) superspace, supervielbeins obey the constraints

T a := dEa = −2iE ∧ σaE , Tα := dEα = 0 , T α := dEα = 0 , (1.19)

which can be solved by

Ea = dxa − idθασaααθα + iθασaααdθα , Eα = dθα , Eα = dθα (1.20)

expressing the supervielbein in terms of superspace coordinates (1.1). Decomposing theexterior derivative on the supervielbein basis

d = EαDα + EαDα + EaDa , (1.21)

4

Chapter 1. Introduction

we obtain the expressions for supersymmetric covariant derivatives,

Da = ∂a , Dα = ∂α + i(σaθ)α∂a , Dα = ∂α + i(θσa)α∂a = −(Dα)∗ . (1.22)

These obey the superalgebra with only one nontrivial (anti-)commutation relation,

Dα, Dα = 2iσaαα∂a , (1.23)

while the other (anti-)commutators vanish

Dα, Dβ = 0 Dα, Dβ = 0

[∂µ, Dα] = 0 = [∂µ, Dα] [∂µ, ∂ν ] = 0 . (1.24)

1.5. Supergravity

Supergravity, a supersymmetric version of Einstein’s gravity, has local supersymmetry asgauge symmetry. As mentioned earlier in this chapter, supersymmetry can be realizedon–shell or off–shell which means the supersymmetry algebra closes with/without using theequations of motion. This also applies to local supersymmetry. There are several differentoff–shell formulations of supergravity which differ by choices of auxiliary field sector. Inthis thesis we will use the so–called minimal supergravity. In chapter 3 we will use minimalN = 1 D = 4 off–shell supergravity in its superspace formulation which we are going todescribe now.

1.5.1. Minimal Supergravity

The list of superspace constraints of minimal supergravity [82, 151–153] is 1

Tαβa = −2iσa

αβ,

TαβA = 0 = Tαβ

A,

Tαβγ = 0,

Tαbc = 0,

Rαβab = 0. (1.25)

With these constraints, the identities (1.17) express the torsion and curvature forms throughthe set of main superfields

Ga := 2i(Taββ − Taβ β) , (1.26)

R := −13Rαβ

αβ = (R)∗ , (1.27)

Wαβγ := 4iσcγγRγcαβ = W (αβγ) = (W αβγ)∗ . (1.28)

1A minimal complete set of superspace constraints for the minimal supergravity multiplet [81] can be found,e.g., in [154–156]; see [91, 157] for a discussion of the algebraic origin of the supergravity constraints.

5

1.5. Supergravity

The final expressions for the superspace torsion 2–forms are,

T a = −2iσaααEα ∧ Eα − 1

8Eb ∧ EcεabcdG

d , (1.29)

Tα = i8Ec ∧ Eβ(σcσd)β

αGd −− i

8Ec ∧ EβεαβσcββR + 1

2Ec ∧ Eb Tbc

α , (1.30)

T α = i8Ec ∧ EβεαβσcββR−

− i8Ec ∧ Eβ(σdσc)

αβ G

d + 12Ec ∧ Eb Tbc

α . (1.31)

The Bianchi identities also imply the following equations for the main superfields,

DαR = 0 , DαR = 0 , (1.32)

DαWαβγ = 0 , DαW αβγ = 0 , (1.33)

DαGαα = DαR , DαGαα = DαR , (1.34)

DγWαβγ = DγD(αGβ)γ ,

DγW αβγ = DγD(α|Gγ|β) . (1.35)

The superspace Riemann curvature 2-form can be decomposed on the anti-self-dual andself-dual parts enclosed in symmetric spin–tensor 2–forms Rαβ and Rαβ = (Rαβ)∗. As aconsequence of the constraints (1.25)

Rαβ ≡ dwαβ − wαγ ∧ wγβ ≡ 14Rab(σaσb)

αβ =

= −12Eα ∧ EβR− i

8Ec ∧ E(α σc

γβ)DγR +

+ i8Ec ∧ Eγ(σcσd)γ

(βDα)Gd −− i

8Ec ∧ EβσcγβW

αβγ + 12Ed ∧ EcRcd

αβ . (1.36)

In our conventions the spinor covariant derivatives are defined by the following decompositionof the covariant differential D

D := EADA = EaDa + EαDα =

= EaDa + EαDα + EαDα . (1.37)

(hence, (Dα)∗ = −Dα). The algebra of covariant derivatives DA, Eq. (1.37), is encoded inthe Ricci identities

DDVA = R BA VB ↔

DDVa = R b

a Vb ,

DDVα = R βα Vβ ,

DDVα = R βα Vβ ,

(1.38)

where VA = (Va, Vα, Vα) is an arbitrary supervector with tangent superspace Lorentz indices.If we decompose them on the basic 2-forms EA ∧ EB, one finds (see [154, 155])

[DA ,DBVC = −TABDDDVC +RAB CD VD . (1.39)

6

Chapter 1. Introduction

When the constraints (1.29), (1.30), (1.31), (1.32), (1.33), (1.34), (1.36) are taken intoaccount, Eqs. (1.38) (or (1.39)) imply

Dα,Dβ Vγ = −Rεγ(αVβ) , (1.40)

Dα,Dβ V γ = −RV(αδβ)γ , (1.41)

Dα, Dβ = 2iσaαβDa ≡ 2iDαβ , etc. (1.42)

The Bianchi identities also allow to find the expression for superfield generalization of thegravitino field strength, Tbc

α(Z),

Tαα ββ γ ≡ σaαασbββεγδTab

δ = −18εαβD(α|Gγ|β) − 1

8εαβ[Wαβγ − 2εγ(αDβ)R] . (1.43)

As a result, the superfield generalization of the left hand side of the supergravity Rarita–Schwinger equation reads

εabcdTbcασdαα =

i

8σaββD(β|Gβ|α) +

3i

8σaβαDβR . (1.44)

The superfield generalization of the Ricci tensor is

Rbcac =

1

32(DβD(α|Gα|β) − DβD(βGα)α)σaαασbββ − (1.45)

− 3

64(DDR +DDR− 4RR)δab .

This suggests that superfield supergravity equation should have the form

Ga = 0 , (1.46)

R = 0 , R = 0 . (1.47)

1.5.2. Superfield supergravity action and admissible variation ofconstrained supervielbein

The superfield action of the minimal off-shell formulation of D = 4, N = 1 supergravity [158]is given by the superdeterminant (or Berezinian) of the matrix of supervielbein coefficients,EAM(Z) in (1.13), which obey the set of supergravity constraints (1.29), (1.30), (1.31),

SSG =

∫d8Z E :=

∫d4xd4θ sdet(EA

M) . (1.48)

One can obtain the supergravity superfield equations (1.46), (1.47) by varying thesuperspace action (1.48). This is not straightforward because the supervielbein superfieldsare restricted by the constraints (1.29), (1.30),(1.31). There are two basic ways to solve thisproblem. The first consists in solving the superspace supergravity constraints in terms ofunconstrained superfields– pre-potentials [89, 151, 153, 159] – the set of which in the caseof minimal supergravity can be restricted to the axial vector superfield Hµ(Z) = Hµ(x, θ, θ)

7

1.5. Supergravity

[151] and the chiral compensator superfield Φ(x+ iH(Z), θ) [159].

Another way, called the Wess–Zumino approach to superfield supergravity [82, 155, 158,160, 161], does not imply to solve the constraints but rather to solve the equations whichappear as a result of requiring that the variations of the supervielbein and spin connection

δE AM(Z) = E B

MKAB (δ) , δwabM(Z) = E C

MuabC (δ) , (1.49)

preserve the superspace supergravity constraints [158], Eqs. (1.29), (1.30), (1.31).

Actually this procedure of finding admissible variations of the supervielbein and superspacespin connection is a linearized but covariantized version of the constraint solution used inpre-potential approach. The independent parameters of admissible variations clearly reflectthe pre-potential structure of the off-shell supergravity. In the case of minimal supergravitytheir set contains [158] (see also [72]) δHa, corresponding to the variation of the axialvector pre-potential of [151], and complex scalar variations δU and δU entering (1.49)only under the action of chiral projectors (as (DD − R)δU and (DD − R)δU) and thuscorresponding to the variations of complex pre-potential of the chiral compensator of theminimal supergravity. The admissible variations of supervielbein read [72, 158]

δEa = Ea(Λ(δ) + Λ(δ))− 1

4Ebσααb [Dα, Dα]δHa + iEαDαδHa − iEαDαδHa ,

(1.50)

δEα = EaΞαa (δ) + EαΛ(δ) +

1

8EαRσaα

αδHa , (1.51)

where

2Λ(δ) + Λ(δ) = 14σααa DαDαδHa + 1

8GaδH

a + 3(DD − R)δU (1.52)

and the explicit expression for Ξαa (δ) in (1.51) can be found in [72]. Neither that nor the

explicit expressions for the admissible variations of spin connection in (1.49) will be neededfor our discussion along this thesis.

Indeed, the variation of superdeterminant of the supervielbein of the minimal supergravitysuperspace can be calculated using (1.50), (1.51) only and reads (see [158])

δE = E[− 1

12σααa [Dα, Dα]δHa +

1

6Ga δH

a +

+2(DD −R)δU + 2(DD − R)δU ] . (1.53)

Taking into account the identity [158]∫d8Z E (DAξA + ξBTBA

A)(−1)A ≡ 0 , (1.54)

one finds the variation of the minimal supergravity action (1.48)

δSSG =∫d8ZE [1

6Ga δH

a − 2R δU − 2R δU ] . (1.55)

8

Chapter 1. Introduction

This clearly produces the superfield supergravity equations of the form (1.46), (1.47) whichresult in the Rarita–Schwinger equation and Einstein equation without cosmological constant

εabcdTbcασdαα = 0 , (1.56)

Rbcac = 0 . (1.57)

9

CHAPTER 2

D = 4 N = 1 SUPERMEMBRANEINTERACTION WITH DYNAMICAL

SCALAR SUPERFIELD

In this chapter we will present the action for the dynamical D = 4 N = 1 supermembranein interaction with a dynamical scalar multiplet. We will show that this action is invariantunder fermionic transformations called κ–symmetry reflecting that its ground state

preserves a part of target space supersymmetry. The possibilty for constructing such an actionis related to the existence of a nontrivial Chevalley-Eilenberg 3–cocycle (i.e a supersymmetricinvariant closed 4–form [162, 163]) constructed from the scalar supermultiplet which entersthe Wess-Zumino part of the action allowing us to couple the D = 4 N = 1 supermembraneto scalar supermultiplet. To this end we will review the well known description of thescalar supermultiplet by chiral superfield in superspace. We present the supermembraneaction in the off–shell scalar supermultiplet background and find the equations of motion.Then we discuss the special scalar supermultiplet contructed from real (instead complex)prepotential, which allows a dual description by 3–form potential C3

′ in flat D = 4 N = 1superspace. Just this special scalar supermultiplet can be coupled to supermembrane (beyondthe background field approximation) as far as the above C3

′ pulled–back to supermembraneworldvolume W 3 serves to construct the Wess–Zumino term of the supermembrane action.

We describe the dynamical system of the special scalar supermultiplet interacting withsupermembrane and obtain the equations of motion with supermembrane source. In thesimplest case in which the scalar multiplet part of the action contains only the simplestkinetic term we also extract the equations of motion for the physical fields. We solve thesedynamical field equations for the physical fields at leading order in supermembrane tension.We conclude this chapter by disscusing the inclusion of nontrivial superpotential and therelation with known domain wall solutions.

11

2.1. Free supermembrane in flat D = 4 N = 1 superspace

2.1. Free supermembrane in flat D = 4 N = 1

superspace

The possibility to construct the D = 4, N = 1 supermembrane action is related to thatin D = 4, N = 1 superspace there exists the following supersymmetric invariant closed4-form1

h4 = dc3 := − i4Eb ∧ Ea ∧ Eα ∧ Eβσabαβ +

i

4Eb ∧ Ea ∧ Eα ∧ Eβσabαβ . (2.1)

This describes a 3-cocycle which is nontrivial in Chevalley-Eilenberg (CE) cohomology[162, 163], which implies that h4 is a supersymmetric invariant closed four form, dh4 = 0and, despite it can be expressed as an exterior derivative of a 3-form, h4 = dc3 (and, hence,is trivial cocycle of de Rahm cohomology), the corresponding 3-form c3 is not invariantunder supersymmetry.

The action for a free supermembrane in D = 4, N = 1 superspace reads [76]

Sp=2 =1

2

∫d3ξ√g −

∫W 3

c3 =

= −1

6

∫W 3

∗Ea ∧ Ea −∫W 3

c3 , (2.2)

where, in the first line g = det(gmn) is the determinant of the induced metric,

gmn = EamηabE

bn , Ea

m := ∂mxa − i∂mθασaαα ˆθα + iθασaαα∂m

ˆθα , (2.3)

W 3 is the supermembrane worldvolume the embedding of which into the target superspace

Σ(4|4) is defined parametrically by the coordinate functions zM(ξ) = (xa(ξ) , θα(ξ), ˆθα(ξ));ξm = (ξ0, ξ1, ξ2) are local coordinates on W 3,

W 3 ⊂ Σ(4|4) : zM = zM(ξ) = (xa(ξ) , θα(ξ), ˆθα(ξ)) . (2.4)

Finally,

c3 :=1

3!Ea3 ∧ Ea2 ∧ Ea1ca1a2a3(Z) =

1

3!dξm3 ∧ dξm2 ∧ dξm1 cm1m2m3 =

= −1

6d3ξεm1m2m3 cm1m2m3 (2.5)

is the pull–back of the 3-form defined in Eq. (2.1) to W 3, so that the second, Wess–Zuminopart of the action can be written in the form of (see [76])

∫W 3

c3 = −16

∫d3ξεm1m2m3 cm1m2m3 .

Here we consider only the case of closed supermembrane so that the worldvolume W 3

1Recall that in our notation the exterior derivative acts from the right, so that for any p-form Ωp andq-form Ωq, d(Ωp ∧ Ωq) = Ωp ∧ dΩq + (−)qdΩp ∧ Ωq. See 1.11

12

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

has no boundary, ∂W 3 = 0/, and∫W 3 d(...) = 0. Then we do not need in the explicit form

of cm1m2m3 in (2.5) as far as variation of its integral in (2.2) can be calculated (using theLie derivative formula, δc3 = iδdc3 + diδc3) through its exterior derivative, the pull–backh4 := h4(Z) of the CE cocycle (2.1), h4 = dc3.

In the second line of Eq. (2.2) we have written the first, Nambu-Goto term of the actionas an integral of a differential three form. This is constructed from the pull–back of thebosonic vielbein form

Ea = dξmEam , Ea

m := ∂mxa − i∂mθασaαα ˆθα + iθασaαα∂m

ˆθα , (2.6)

using the worldvolume Hodge star operation,

∗ Ea :=1

2dξm ∧ dξn√gεmnkgklEa

l . (2.7)

The action (2.2) is invariant under the local fermionic κ–symmetry transformations. Thesehave the form of

δκxµ = iκασµααθ

α − iθασµαακα , δκθα = κα , δκθ

α = κα , (2.8)

where the spinorial fermionic parameter κα = κα(ξ) = (κα)∗ has actually only two indepen-dent components because it obeys the equations

κα = κβγβα ⇔ κα = κα ˜γβα (2.9)

with

γβα = εβαεαβ ˜γβα =i

3!√gσaβαεabcdε

mnkEbmE

cnE

dk . (2.10)

By construction, the matrix γ obeys

γββ ˜γβα = δβα (2.11)

which makes two equations in (2.9) equivalent.To prove the κ–symmetry one has to use the identities

1

2Ec ∧ Eb ∧ Eασbcαβ = ∗Ea ∧ Eα(σa ˜γ)αβ (2.12)

which allows to present the variation of the kinetic, Nambu-Goto type, and the Wess–Zuminoterms in similar form.

It is convenient to write the κ–symmetry transformations in the form of

iκEa := δκZ

MEMa(Z) = 0 ,

iκE

α := δκZMEM

α(Z) = κα = κα ˜γβα ,

iκˆEα := δκZ

MEMα(Z) = κα = κβγβα .

(2.13)

13

2.2. Scalar supermultiplet as described by chiral superfield

2.2. Scalar supermultiplet as described by chiralsuperfield

In this section we review the well known description of scalar supermultiplet by chiralsuperfield in superspace [154, 155, 164].

As noted in section 1.2 superfields are highly reducible representations of the supersym-metry algebra but it is possible to extract irreducible representations from them by imposingsuitable covariant constraints. The simplest irreducible representation of the D = 4, N = 1supersymmetry, the scalar supermultiplet, is described by the chiral superfield, this is to sayby complex superfield obeying the so-called chirality equation

DαΦ = 0 . (2.14)

The complex conjugate (c.c.), Φ = (Φ)∗, obeys

DαΦ = 0 (2.15)

and is called anti–chiral superfield. The free equations of motion for the physical fields of amassless scalar supermultiplet (φ(x) = Φ|θ=0 and iψα(x) = DαΦ|θ=0) are collected in thesuperfield equation

DDΦ := DαDαΦ = 0 . (2.16)

This equation and its c.c. can be derived from the action

Skin =

∫d8zΦΦ , (2.17)

where the superspace integration measure d8z = d4xd2θd2θ is normalized as2

d8z = d4xDD DD := d4xDαDα DαDα . (2.18)

Indeed, the variation of this functional reads δSkin =∫d8z (ΦδΦ + δΦ Φ). As far as the

variation of chiral superfield should be chiral, DαδΦ = 0, and DαδΦ = 0, we can equivalentlywrite the action variation as δSkin =

∫d4x DαD

α((DDΦ)δΦ) + c.c., which results in theequations of motion (2.16).

The most general selfinteraction of the scalar supermultiplet is described by the superfieldaction

Ss−int[Φ; Φ] =

∫d8zK(Φ, Φ) +

∫d6ζLW (Φ) +

∫d6ζR W (Φ) =

=

∫d4x DD DD K(Φ, Φ) +

∫d4xDDW (Φ) +

∫d4x DD W (Φ) ,(2.19)

2Notice that, although the r.h.s. of this equation is not manifestly hermitian, its imaginary part is integralof complete derivative (as far as DD DD = DDDD −4iσa αα∂a[Dα, Dα]) and, as such, can be ignoredin our discussion.

14

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

where K(Φ, Φ) is an arbitrary function of chiral superfield and its complex conjugate calledKahler potential and W (Φ) (= (W (Φ)∗) is an arbitrary holomorphic function of the complexscalar superfield Φ called superpotential. This latter is chiral, DαW (Φ) = W ′(Φ)DαΦ = 0,and hence is integrated with chiral measure defined by d6ζL = d4xDD (and d6ζR = d4xDD).To have the standard kinetic term for the scalar field of the supermultiplet, the Kahlerpotential is usually chosen to obey

K′′ϕ ϕ(ϕ, ϕ) :=∂

∂ϕ

∂

∂ϕK(ϕ, ϕ) 6= 0 . (2.20)

The superfield equations of motion following from the action Ss−int[Φ; Φ] (2.19) are

E := DDK′Φ + W ′Φ =

= DDΦ K′′ΦΦ(Φ, Φ) +DαΦDαΦ K′′′ΦΦΦ(Φ, Φ) + W ′Φ(Φ) = 0 , (2.21)

E := DDK′Φ +W ′Φ =

= DDΦ K′′ΦΦ(Φ, Φ) + DαΦ DαΦ K′′′ΦΦΦ(Φ, Φ) +W ′Φ(Φ) = 0 , (2.22)

where prime denotes the derivative with respect to argument, K′Φ

:= ∂K(Φ,Φ)

∂Φ, K′′

ΦΦ:=

∂2

∂Φ2K(Φ, Φ), etc. These equations can be obtained by solving the chirality conditions (2.14)and (2.15) in terms of prepotential, generic complex superfield P (= (P )∗),

Φ = DDP , Φ = DDP , (2.23)

and vary with respect to this prepotential and its complex conjugate,

E =δSs−int[Φ; Φ]

δP, E =

δSs−int[Φ; Φ]

δP. (2.24)

2.2.1. Four form field strength constructed from the scalarsupermultiplet

Having a chiral superfield K,

DαK = 0 , DαK = 0 , (2.25)

one can construct the following supersymmetric invariant closed four form (CE cocycle) inflat D = 4, N = 1 superspace [83]

F4 = dC3 :=1

4Eb ∧ Ea ∧ Eα ∧ Eβσabαβ K +

1

4Eb ∧ Ea ∧ Eα ∧ Eβσabαβ K +

+1

4!Ec ∧ Eb ∧ Ea ∧ Eαεabcdσ

dαβDβK +

1

4!Ec ∧ Eb ∧ Ea ∧ Eβεabcdσ

dαβDαK −

+1

4!Ed ∧ Ec ∧ Eb ∧ Ea i

4εabcd

(DDK −DDK

). (2.26)

15

2.3. Supermembrane action in the scalar multiplet background

Notice that we intentionally have not used the notation Φ for chiral superfield to stress that,e.g. having a free chiral superfield of Eqs. (2.14) satisfying equations of motion (2.16), onecan construct the three form using some holomorphic functions K = K(Φ), K = K(Φ)which obey DDK = K ′′(Φ)DαΦDαΦ instead of (2.16).

Interestingly enough, F4 in (2.26) can be considered as real part of the complex closedform FL4 ,

F4 = <e(FL4 ) := 12

(FL4 + FR4

), (2.27)

FL4 = 14Eb ∧ Ea ∧ Eα ∧ Eβ σab αβ K + 1

4!Ec ∧ Eb ∧ Ea ∧ Eα εabcdσ

dαβDβK +

+ 14!Ed ∧ Ec ∧ Eb ∧ Ea i

16εabcd DDK , DαK = 0 , (2.28)

FR4 = 14Eb ∧ Ea ∧ Eα ∧ Eβ σab αβK + 1

4!Ec ∧ Eb ∧ Ea ∧ Eα εabcdσ

dβαD

βK +

+ 14!Ed ∧ Ec ∧ Eb ∧ Ea i

16εabcdDDK , DαK = 0 . (2.29)

The fact that these forms are closed as a consequences of (2.25),

dFL4 = 0 ⇔ DAK = 0 , (2.30)

dFR4 = 0 ⇔ DAK = 0 , (2.31)

suggests the existence of the complex 3-form potentials CL3 and CR

3 = (CL3 )∗ such that

FL4 = dCL3 and FR4 = dCR

3 .

To study a supermembrane in the background of scalar multiplet, which will be the subjectof the next section, we do not need an explicit expression for C3. However, we do need it toobtain the equations for the scalar multiplet fields with a source from supermembrane, sothat we will come back to discussing the problem of constructing potentials in section 2.4.

2.3. Supermembrane action in the scalar multipletbackground

The action of supermembrane in the background of a scalar multiplet can be written in theform

Sp=2 =1

2

∫d3ξ√KK√g −

∫W 3

C3 ,

= −1

6

∫W 3

∗Ea ∧ Ea√KK −

∫W 3

C3 , (2.32)

where C3 is the pull–back of the C3 potential defined by Eq. (2.26) involving the chiralsuperfields K and K (2.25). For simplify we omit the hat symbol from the pull–backs ofsuperfields here and below in the places where this cannot produce a confusion.

16

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

The action (2.32) is invariant under the κ–symmetry transformations

iκEa = 0 , iκE

α = κα , iκˆEα = κα , (2.33)

with iκdZM := δκZ

M , similar to ones in (2.13) and (2.8) but with the spinorial parameterobeying the reducibility conditions

κα = −iκβγβα√K/K ⇔ κα = iκα ˜γβα

√K/K (2.34)

defined by a projector which differs from the one in (2.9) by a (super)field dependent phase

factor i√K/K .

Notice that, if we write the counterpart of the action (2.32) with an arbitrary function

S(K, K) instead of√KK and perform the fermionic variation (2.33) of such an action, we

find that the local fermionic κ–symmetry parameter should obey the equations κα∂S/∂K =

i/2κβγβα and κβ∂S/∂K = καγαβ. This system of equations has a nontrivial solution

when ∂S/∂K = 14∂S/∂K . This latter equation is solved by S(K, K) =

√KK so that the

action (2.32) for scalar multiplet in supergravity background can be constructed from therequirement of the κ–symmetry.

2.3.1. Equations of motion for supermembrane in a background ofan off–shell scalar supermultiplet

The supermembrane equations of motion can be obtained by varying the action (2.32)with respect to coordinate functions ZM(ξ), so that we can write them in the form ofδSp=2

δZM (ξ)= 0. The convenient form of the bosonic and fermionic equations can be extracted

by multiplying this on the inverse supervielbein, EMα (Z) δSp=2

δZM (ξ)= 0 and EM

a (Z) δSp=2

δZM (ξ)= 0.

These combinations appear as the coefficients for iδEα := δZM(ξ)EM

α(Z), iδˆEα :=

δZM(ξ)EMα(Z) and iδE

a := δZM(ξ)EMa(Z) in the integrand of the action variation.

This implies the possibility to write the formal expression for supermembrane equations ofmotion in the form

δSp=2

iδEα:= EM

α (Z)δSp=2

δZM(ξ)= 0 ,

δSp=2

iδˆEα

:= EMα (Z)

δSp=2

δZM(ξ)= 0 , (2.35)

δSp=2

iδEa:= EM

a (Z)δSp=2

δZM(ξ)= 0 . (2.36)

17

2.4. Superfield equations for the dynamical system of special scalar supermultipletinteracting with supermembrane

The straightforward calculation gives the following explicit form of these equations of motion

∗Ea ∧(i ˆEασaαα

√KK − Eβ(γσa)α

βK)

+1

12∗ Ea ∧ Ea

(√K/KDαK − iγααDαK

)= 0 ,

(2.37)

∗Ea ∧(iEασaαα

√KK − ˆEβ(σaγ)β αK

)+

1

12∗ Ea ∧ Ea

(√K/KDαK − iDαKγαα

)= 0 ,

(2.38)

D(∗Ea) −1

6∗ Eb ∧ Eb

(Da ln K +Da lnK

)+

1

2∗ Ea ∧ (d ln ˆK + d ln K)−

− i

12√KK

Ed ∧ Ec ∧ Eb εabcd (DDK −DDK)−

− 1

4√KK

Ec ∧ Eb ∧ εabcdσdαα (EαDαK + ˆEαDαK)−

−Eb ∧ Eα ∧ Eβσabαβ

√K/K − Eb ∧ ˆEα ∧ ˆEβσabαβ

√K/K = 0 .(2.39)

Notice that the above equations of motion are not independent. According to the secondNoether theorem, the gauge symmetries of a dynamical system result in the so-calledNoether identities relating the left-hand sides of equations of motion of this system. Thesupermembrane possesses a number of gauge symmetries, including the local fermionicκ–symmetry (2.33), (2.34). This is reflected by the fact that contracting our fermionic

equation (2.37) with i√K/K ˜γβα we arrive at Eq. (2.38). Denoting the left hand sides of

equations (2.37) and (2.38) by Ψα and Ψα, respectively, we can write the above describedNoether identity for the κ–symmetry in the form of

˜γβαΨα ≡ −i√K/Kε

˙βαΨα . (2.40)

2.4. Superfield equations for the dynamical system ofspecial scalar supermultiplet interacting withsupermembrane

2.4.1. Special scalar multiplet and its dual three form potential

In our discussion below we will be considering not generic but special scalar multipletdescribed by the chiral superfield constructed from the real prepotential V = (V )∗,

Φ = DDV , Φ = DDV . (2.41)

18

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

On the level of auxiliary fields the distinction of this special case is that one of the realauxiliary scalars of the generic scalar multiplet is replaced in it by a divergence of a realvector, ∂µk

µ or, equivalently, by the field strength of a three form potential kνρσ = kµεµνρσ(in this latter form it was described in [83] and, as one of ’variant superfield representations’,in [86]).

Indeed, the complex prepotential P of the generic chiral multiplet, Φ = DDP , Φ = DDP ,is defined up to the gauge transformations, P 7→ P + DαΞα. These imply that theimaginary part of the generic prepotential is transformed by =mP := (P − P )/2i 7→=mP + (DαΞα − DαΞα)/2i. Hence not-pure gauge parts of the superfield parameter=mP are the ones which do not have their exact counterparts in the composed superfield(DαΞα −DαΞα)/2i. One can check that the superfield parameter DαΞα −DαΞα has allthe components but one having contributions of different independent functions withoutderivatives. The only exception is the highest component in its decomposition which reads−4iθθ θθ∂a(k

a + ka) and includes the divergence of the real part of the complex vectorka = σααa (DαΞα)|θ=0 = (ka)

∗ versus an arbitrary function in a generic real scalar superfield,like P . Then one can guess that the equations of motion for the special scalar supermultipletwill differ from the set of equations for a generic scalar supermultiplet by that one of thealgebraic auxiliary field equations of the latter ((E−E)|0 = 0) will be replaced by its derivative(∂a(E − E)|0 = 0). In other words, the general solution of the (auxiliary) field equations ofthe special scalar supermultiplet involves one additional (with respect to the generic case)arbitrary real constant. Furthermore, this indicates that the above mentioned auxiliary fieldequation of the special scalar supermultiplet (∂a(E − E)|0 = 0) is dependent, i.e. can beobtained as a consequence of other equations; this implies that the only effect of the use ofthe complex prepotential in the generic case is vanishing of a real constant which is indefinitein the case of special scalar multiplet (where ∂a(E − E)|0 = 0 ⇒ (E − E)|0 = −2ic). Wewill see that this is indeed the case.

Equations of motion of special scalar multiplet

The variation of the general action (2.19) for the special chiral superfields (2.41) with respectto real prepotential V apparently produces only the real part of the complex equation (2.22),

δSs−int[DDV ;DDV ]

δV= 0 ⇒ E + E := DDK′Φ +DDK′Φ +W ′

Φ + W ′Φ = 0 .(2.42)

However, as far as the left hand sides of the equations of motion for generic scalarmultiplet, Eqs. (2.22) and (2.21), are, respectively, anti-chiral and chiral, DαE = 0 andDαE = 0, Eq. (2.42) implies that the imaginary part of the complex equation (2.22) isequal to a constant,

E + E = 0 ⇒ Dα(E − E) = 0 ⇒ ∂a(E − E) = 0 . (2.43)

Hence the only effect of the use of the special chiral superfields (2.41) instead of the genericscalar superfield (2.23) is that the equation E = 0 is replaced by E = −ic where c is anarbitrary real constant.

19

2.4. Superfield equations for the dynamical system of special scalar supermultipletinteracting with supermembrane

On spontaneous supersymmetry breaking

The presence of this arbitrary constant in the right hand side of the superfield equations ofmotion, E = −ic, actually suggests a possible spontaneous supersymmetry breaking in thetheory of special chiral multiplet. To clarify this, let us discuss the simple case of a freemassless special scalar multiplet, in which the action reads

∫d8zΦΦ =

∫d8zDDV DDV

so that E = DDΦ and the equations of motion (2.42) simplify to

DDΦ + DDΦ = 0 . (2.44)

As it has been discussed above (Eq. (2.43)) these equations lead to Dα(DDΦ− DDΦ) = 0and ∂a(DDΦ− DDΦ) = 0. Algebraically all this set of equations is solved by DDΦ = −icwith the above mentioned arbitrary real constant c,

DDΦ + DDΦ = 0 ⇒ DDΦ = −ic , c = const . (2.45)

In particular, this constant enters the solution of auxiliary field equations which now reads

DDΦ|0 = −ic , c = const . (2.46)

As the on-shell supersymmetry transformations of the fermionic fields ψα = −iDαΦ|0 areobtained from the off-shell ones, δψα = i

2εβDDΦ|0 + 2(σaε)α∂aφ, by inserting the above

solution of the auxiliary field equations, they read

δψα =c

2εβ + 2(σaε)α∂aφ . (2.47)

Hence, for nonvanishing value of c, the on-shell supersymmetry transformations of δψαcontains the additive contribution of supersymmetry parameter εβ characteristic of thetransformation rules of the Volkov-Akulov Goldstone fermion [15, 16] the presence of whichmay be considered as an indication of the spontaneous supersymmetry breaking.

However, studying more carefully the case of free special scalar multiplet, one findsthat such a spontaneous symmetry breaking actually does not occur if nontrivial boundaryconditions are not introduced. Indeed, the constant in the superfield equations (2.45) can bereproduced from the generic scalar supermultiplet action which includes the superpotentiallinear in chiral superfield, W (Φ) = −icΦ. As it was observed already in [165], such aterm can be removed from the action by a field redefinition. However, the boundary termcontribution may change the situation; this role can be also played by supermembranecontribution. Further discussion on spontaneous supersymmetry breaking in the interactingsystem of scalar multiplet and supermembrane goes beyond the scope of this thesis. Weturn to the three form potential presentation of the special chiral supermultiplet.

20

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

Dual three form potential

The four form field strength constructed with the use of special scalar multiplet (2.41) isobtained from (2.26) by substituting

K = Φ = DDV , K = Φ = DDV . (2.48)

It reads

F4 = dC ′3 = 14Eb ∧ Ea ∧ Eα ∧ Eβ σabαβDDV + 1

4Eb ∧ Ea ∧ Eα ∧ Eβ σab αβDDV +

+ 14!Ec ∧ Eb ∧ Ea ∧

(Eα εabcdσ

dαβDβDDV + Eα εabcdσ

dβαD

βDDV)

+

+ 14!Ed ∧ Ec ∧ Eb ∧ Ea i

4εabcd (DDDDV −DDDDV ) . (2.49)

The corresponding 3-form potential C ′3 can be written in terms of the real prepotential asfollows [83]

C ′3 = 2iEc ∧ Eα ∧ Eβ σc αβ V +1

2Ec ∧ Eb ∧ Eασbc α

βDβV −

−1

2Eb ∧ Ea ∧ Eβ σab

αβ DαV −

1

4!Ec ∧ Eb ∧ Eaεabcdσ

d βα [Dα , Dβ]V . (2.50)

Of course, this expression can be changed on an equivalent one using gauge transformationsδC3 = dα2. These do not change the field strength (2.49) and are responsible for thepossibility to do not have the lower dimensional form contributions (∝ Eα ∧ Eβ ∧ Eγ etc.)in the above C ′3.

The existence of this simple three form C ′3 giving a dual description of the special chiralsupermultiplet (2.41) is the main reason to restrict our discussion below by this special case.

2.4.2. Superfield equations of motion for interacting system

Let us consider the most general interaction of the special scalar supermultiplet withsupermembrane as described by the action (2.32) with K = Φ = DαD

αV and K = Φ =DαDαV as in (2.48), i.e. by

S =

∫d8zK(Φ, Φ) +

∫d6ζLW (Φ) + c.c.+

1

2

∫d3ξ√g

√Φ ˆΦ−

∫W 3

C ′3 = (2.51)

=

∫d4x DDDDK(Φ, Φ) +

∫d4x(DDW (Φ) + c.c.)− 1

6

∫W 3

∗Ea ∧ Ea

√Φ ˆΦ−

∫W 3

C ′3

with special chiral superfield (2.41),

Φ = DαDαV , Φ = DαDαV . (2.52)

The variation of the interacting action (2.51) with respect to supermembrane variables

21

2.4. Superfield equations for the dynamical system of special scalar supermultipletinteracting with supermembrane

gives formally the same equations of motion as for the supermembrane in the background,(2.37)–(2.39), but with K = Φ = DDV , 3

∗Ea ∧(i ˆEασaαα

√Φ ˆΦ− Eβ(γσa)α

β ˆΦ

)+

1

12∗ Ea ∧ Ea

(√ˆΦ/Φ DαΦ− iγααDαΦ

)= 0 ,

(2.53)

∗Ea ∧(iEασaαα

√ΦΦ− ˆEβ(σaγ)β αΦ

)+

1

12∗ Ea ∧ Ea

(√Φ/ΦDαΦ− iDαΦγαα

)= 0 ,

(2.54)

D(∗Ea) −1

6∗ Eb ∧ Eb

(Da ln Φ +Da ln Φ

)+

1

2∗ Ea ∧ (d ln Φ + d ln Φ)−

− i

12√

ΦΦEd ∧ Ec ∧ Eb εabcd (DDΦ−DDΦ)−

− 1

4√

ΦΦEc ∧ Eb ∧ εabcdσdαα (EαDαΦ + ˆEαDαΦ)−

−Eb ∧ Eα ∧ Eβσabαβ

√Φ/Φ− Eb ∧ ˆEα ∧ ˆEβσabαβ

√Φ/Φ = 0 . (2.55)

However, the target superspace superfields the pull–backs of which enter these equations haveto be the solutions of interacting equations with the source terms from the supermembrane.These superfield interacting equations read

E + E = J , (2.56)

where

E = DDΦ K′′ΦΦ(Φ, Φ) + DαΦ DαΦ K′′′ΦΦΦ(Φ, Φ) +W ′Φ(Φ)

(2.57)

(see (2.22) and (2.42)) and

J(z) = − δSp=2

δV (z)(2.58)

is the current superfield from the supermembrane. The problem of obtaining the completeset of interacting equations for the dynamical system of supermembrane and special chiralsupermultiplet is now reduced to the problem of calculating this supermembrane current.

3To simplify the expressions, we omitted the hat symbol from the pull–backs of superfields and theirderivatives in equations (2.54) and (2.55), but, in contrast, left all the pull–back symbols in equation(2.53) so that one can appreciate simplification comparing this with its complex conjugate Eq. (2.54).

22

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

2.4.3. Supermembrane current

The supermembrane current is split naturally on the contributions from the Nambu–Gotoand the Wess–Zumino terms of the action (2.32) with (2.41)

J(z) = JNG(z) + JWZ(z) = − δSp=2

δV (z)(2.59)

The Nambu-Goto part of the current

JNG(z) := − δδV (z)

12

∫d3ξ√g

√DDV DDV (2.60)

(DDV := DαDαV (z)|zM=zM (ξ)) is calculated by first using the properties of the superspacedelta function

δ8(z) :=1

16δ4(x) θθ θθ ,

∫d8z δ8(z − z′)f(z) = f(z′) (2.61)

to present (2.60) in the form

JNG(z) = − δδV (Z)

12

∫d8z′

√DDV (z′)DDV (z′)

∫d3ξ√gδ8(z′ − z) . (2.62)

Then the calculation reduces to using the definition of variation δV (z′)δV (z)

= δ8(z′ − z) andperforming the superspace integration. In such a way one arrives at

JNG(Z) = −1

4

∫d3ξ√g

√Φ

ˆΦDDδ8(z − z)− 1

4

∫d3ξ√g

√ˆΦ

ΦDDδ8(z − z) ,(2.63)

where Φ = DDV , Φ = DDV (see Eqs. (2.23)) and Φ := Φ(z(ξ)) etc. Similarly one canpresent the Wess–Zumino current in the form of

JWZ(Z) =

(2i

∫W 3

Ec ∧ Eα ∧ Eασcαα+

+1

2

∫W 3

Ec ∧ Eb ∧ EασbcαβDβ −

1

2

∫W 3

Ec ∧ Eb ∧ EασbcβαDβ−

− 1

4!

∫W 3

Ec ∧ Eb ∧ Eaεabcdσdαα[Dα, Dα]

)δ8(z − z) . (2.64)

2.5. Simplest equations of motion for spacetime fieldsinteracting with dynamical supermembrane

Having the superfield equations with supermembrane current contributions, the next stageis to extract the equations of motion for the physical fields of the supermultiplet. We willdo this for the simplest case when the special scalar multiplet part of the interacting action

23

2.5. Simplest equations of motion for spacetime fields interacting with dynamicalsupermembrane

is given by the kinetic term (2.19) only, this is to say for the interacting system describedby the action

S = Skin + Sp=2 =

∫d8zΦΦ +

1

2

∫d3ξ

√Φ ˆΦ√g −

∫W 3

C3′ (2.65)

where Φ = DDV , Φ = DDV (2.52) and C ′3 is the pull–back to W 3 of the 3-form C ′3defined in (2.50). The interacting equations of motion for the bulk superfields, Eqs. (2.57),in this case simplifies to

DDΦ + DDΦ = J(z) (2.66)

where the current J(z) is given by (2.59), (2.63) and (2.64).

2.5.1. General structure of the simplest special scalar multipletequations with a superfield source

Superfield equation (2.66) encodes the dynamical equations for the physical fields of thescalar multiplet, φ(x) = Φ|0 and ψα(x) = −i(DαΦ)|0, as well as algebraic equations forauxiliary fields DDΦ|0 and DDΦ|0. These latter include the leading component of the realsuperfield equation (2.66)

DDΦ|0 + DDΦ|0 = J(z)|0 (2.67)

as well as the first order equation

∂a(DDΦ|0 − DDΦ|0) = − i4σααa [Dα , Dα]J(z)|0 . (2.68)

The set of dynamical field equations include the Dirac (actually Weyl) equation with thesource from supermembrane,

σaαα∂aψα := −i∂ααDαΦ|0 =

1

4DαJ(z)|0 , (2.69)

and the Klein-Gordon equation, also with the source,

φ(x) := Φ|0 = − 1

16DDJ(z)|0 . (2.70)

Now, to specify supermembrane contributions to the scalar multiplet field equations we haveto calculate the derivatives of the supermembrane current.

2.5.2. Dynamical scalar multiplet equations with supermembranesource contributions

24

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

Auxiliary field equations

The leading components J |0 of the current J in (2.67) is the sum of

JNG|0 =1

16

√φ

φ

∫d3ξ√g ˆθ ˆθδ4(x− x) +

1

16

√φ

φ

∫d3ξ√g θθδ4(x− x) (2.71)

and

JWZ(Z)|0 =1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcdθσd ˆθ δ4(x− x) +O(f 4) , (2.72)

where O(f 4) denotes the terms of the fourth order in fermions (in this case, these are

worldvolume fermionic fields θ, ˆθ and their worldvolume derivatives, ∂mθ := ∂θ/∂ξm andc.c.); the explicit form of these one can find in the Appendix B (Eq. (B.10)).

Substituting the above expressions into Eq. (2.67), one finds that the real part of theauxiliary fields of the chiral multiplet has quite a complex form in terms of supermembranevariables

DDΦ|0 + DDΦ|0 =1

16

√φ

φ

∫d3ξ√g ˆθ ˆθδ4(x− x) +

1

16

√φ

φ

∫d3ξ√g θθδ4(x− x) +

+1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcdθσd ˆθ δ4(x− x) +O(f 4) , (2.73)

where O(f 4) are the same as in Eq. (2.72) (and thus can be read off Eq. (B.10)).

The second auxiliary field equation, Eq. (2.68), reads

∂a(DDΦ|0 − DDΦ|0) = − i

8 · 4!

∫W 3

Ed ∧ Ec ∧ Ebεabcdδ4(x− x) +O(f 2) , (2.74)

where the terms of higher order in fermions, O(f 2) can be found in Eqs. (B.18) and (B.19)of Appendix B (multiplying the expressions presented there by − i

4σααa ). On the first look it

might seem that Eq. (2.74) imposes additional restrictions on the supermembrane motion.Such possible restrictions might come from the selfconsistency condition of Eq. (2.74); atzero order in fermions that reads 4

∂[aεb]c1c2c3

∫W 3

dxc3 ∧ dxc2 ∧ dxc1δ4(x− x) = 0 . (2.75)

However, one can check that this equation is satisfied identically. Indeed, using theidentity εc1c2c3[a∂b] ≡ −3

2εab[c1c2∂c3] one can write the l.h.s. of Eq. (2.75) in the form

of −32εabc1c2

∫W 3

dxc2 ∧ dxc1 ∧ dδ4(x − x) = −32εabc1c2

∫W 3

d (dxc2 ∧ dxc1 δ4(x− x)) which

4∫W 3

Ed ∧ Ec ∧ Ebεabcdδ4(x− x) =∫W 3

dxd ∧ dxc ∧ dxbεabcdδ4(x− x) + fermionic contributions.

25

2.5. Simplest equations of motion for spacetime fields interacting with dynamicalsupermembrane

vanishes as an integral of total derivative in the case of closed supermembrane which we arestudying here (∂W 3 = 0/ ⇒

∫W 3 d(...) = 0) .

As we have discussed in section 2.4.1, the on-shell transformations are obtained from theoff-shell ones, δψα = 2(σaε)α∂aφ+ i

2εβDDΦ|0 for the case of fermions, by substituting the

solution of the equations for the auxiliary fields. This implies that the on-shell supersymmetrytransformation of fermions will be quite complicated due to the complicated structure ofthe auxiliary field equations (2.73) and (2.68). As the on-shell fermionic supersymmetrytransformations can be used to extract BPS conditions for the supersymmetric solutions, theirfurther study in our simple system might lead to useful suggestions for the investigation ofthe backreaction of D=10,11 super-p-branes on the BPS solutions of supergravity equations.

Dynamical field equations

Fortunately, the dynamical equations for the physical fields of the special scalar multipletfollowing from the simplest interacting action Eq. (2.65) do not obtain contributions fromthe auxiliary fields of the scalar multiplet, which on the mass shell are expressed by quitecomplicated Eqs. (2.73) and (2.68).

Specifying the current contributions to (2.69) and (2.70), we find the massless Diracequation with the supermembrane contributions,

σaαα∂aψα =

1

32

√φ

φ

∫d3ξ√g ˆθαδ

4(x− x)−

− 1

8 · 4!

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θσd)α δ

4(x− x) +O(f 3) , (2.76)

and the Klein-Gordon equation, also with the source from supermembrane,

φ(x) =1

64

√φ

φ

∫d3ξ√gδ4(x− x)−

− 1

64

∫W 3

Ec ∧ Eb ∧ dθσbcθ δ4(x− x)− i

64

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θ)2∂dδ4(x− x) +

+i

64

∫d3ξ√g(θσa ˆθ)

√φ

ˆφ∂aδ

4(x− x) +O(f 4) . (2.77)

The explicit form of the terms of higher order in fermions, O(f 4) in (2.77) and O(f 3) in(2.76), can be extracted from the Eqs. (B.7) and (B.16) in Appendix B.

26

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

2.5.3. Simplest solution of the dynamical equations at leadingorder in supermembrane tension

The above equations can be formally solved by

ψα = ψα0 +1

32

∫d3ξ√g

√φ

ˆφ(ˆθσa)α∂aG0(x− x) +

+1

8 · 4!

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θσdσa)α∂aG0(x− x) +O(f 3) , (2.78)

φ(x) = φ0(x) +1

64

∫d3ξ√g

√φ

ˆφ

(G0(x− x) + i(θσa ˆθ)∂aG0(x− x)

)−

− 1

64

∫W 3

Ec ∧ Eb ∧ dθσbcθG0(x− x)− i

64

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θ)2∂dG0(x− x) +

+O(f 4) , (2.79)

where ψα0 and φ0(x) are solutions of the free equations and G0(x− x) is the Green functionof the free D = 4 Klein-Gordon operator := ∂a∂

a,

φ0(x) = 0 , σaαα∂aψα0 = 0 G0(x− x) = δ4(x− x) . (2.80)

Eqs. (2.78) and (2.79) give only formal solutions as far as the pull–back of the phase of the

complex scalar superfield enters their r.h.s. through

√φˆφ

multipliers in the integrants.

Assuming the solution of the homogeneous equation to be real, φ0(x) = (φ0(x))∗

one can solve Eqs. (2.76) and (2.77) in the first order in the supermembrane tensionT (this is set to unity in our equations above and below, but can be easily restored by∫d3ξ√g 7→ T

∫d3ξ√g and

∫W 3

7→ T∫W 3

). This reads (setting back T = 1)

ψα = ψα0 +1

32

∫d3ξ√g (ˆθσa)α∂aG0(x− x) +

+1

8 · 4!

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θσdσa)α∂aG0(x− x) +O(f 3) , (2.81)

φ(x) = φ0(x) +1

64

∫d3ξ√g(G0(x− x) + i(θσa ˆθ)∂aG0(x− x)

)+

− 1

64

∫W 3

Ec ∧ Eb ∧ dθσbcθG0(x− x)− i

64

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θ)2∂dG0(x− x) +

+O(f 4) , φ0(x) = (φ0(x))∗ . (2.82)

The contribution of higher order in string tension would include the product of distributions(of the type G0(x− x(ξ1)) δ4(x− x(ξ2))) and their accounting requires a careful study of aclassical counterpart of the renormalization procedure, similar to the one developed for the

27

2.5. Simplest equations of motion for spacetime fields interacting with dynamicalsupermembrane

radiation reaction problem [166] and for general relativity [167]. The generalization of sucha technique for the case of p–brane has been developed in very recent [168].

2.5.4. Superfield equations with nontrivial superpotential

As a first stage in searching for solution of the superfield equations with a nontrivialsuperpotential, let us consider the relation with known domain wall solutions of the Wess–Zumino model [169, 170]. To this end let us consider our dynamical system with nontrivialsuperpotential and the simplest kinetic term. This is described by the interacting action

S =

∫d8zΦΦ +

∫d6ζLW (Φ) +

∫d6ζRW (Φ) +

1

2

∫d3ξ

√Φ ˆΦ√g −

∫W 3

C3′ . (2.83)

with Φ and Φ and C3′ expressed in terms of real pre-potential V (z) by Eqs. (2.41) and

(2.50).

The superfield equations of motion (2.66) acquire now the superpotential contributions,

DDΦ + DDΦ +W ′Φ(Φ) + W ′

Φ(Φ) = J(z) (2.84)

The auxiliary field equations read

DDΦ|0 + DDΦ|0 +W ′φ(φ) + W ′

φ(φ) = J(z)|0 , (2.85)

∂a(DDΦ|0 + W ′φ(φ)− DDΦ|0 −W ′

φ(φ)) = − i4σααa [Dα , Dα]J(z)|0 , (2.86)

and the dynamical field equations are

σaαα∂aψα +

i

4ψα W

′′φφ(φ) =

1

4DαJ(z)|0 , (2.87)

φ(x)− 1

16DDΦ|0W ′′

φφ(φ) +1

16ψαψ

αW ′′′φφφ(φ) = − 1

16DDJ(z)|0 (2.88)

with the same supermembrane current (2.59), (2.63), (2.64).

In the absence of supermembrane current, J = 0, the auxiliary field equations are solvedby DDΦ|0 = −W ′

φ(φ) + ic, DDΦ|0 = −W ′

φ(φ)− ic and the constant c can be removed

by redefining superpotential W ′φ(φ) 7→ W ′

φ(φ) + ic. Thus, without lost of generality, one

can simplify notation and substitute −W ′φ(φ) for DDΦ|0 in the dynamical equations (2.88).

Domain wall ansatz of [169, 170] implies that all the fields are static and depend on onlyone spatial coordinate which we chose to be x2 = y. Then Eqs. (2.87) and (2.88) withJ = 0 becomes

σ2αα∂yψ

α(y) +i

4ψα(y) W ′′

φφ(φ(y)) = 0 , (2.89)

∂2yφ(y)− 1

16W ′φ(φ)W ′′

φφ(φ)− 1

16ψαψ

αW ′′′φφφ(φ) = 0 (2.90)

28

Chapter 2. D = 4 N = 1 Supermembrane interaction with dynamical Scalar Superfield

Notice that Eq. (2.89) split into the pair of equations for ψ1, ψ1 and ψ2, ψ2,

∂yψ1(y) +1

4ψ1(y) W ′′

φφ(φ(y)) = 0 , ∂yψ2(y) +1

4ψ2(y) W ′′

φφ(φ(y)) = 0 , (2.91)

such that the solution of the second can be constructed from the solution of the first asψ2 = ψ1, ψ2 = ψ1. For such a solution of the fermionic equation ψαψ

α = 0 and the bosonicequation simplifies to ∂2

yφ(y) − 116W ′φ(φ)W ′′

φφ(φ) = 0. This, in its turn, is solved by any

solution of the following first order BPS equations [169, 170]

∂yφ(y)− eiα

4W ′φ(φ(y)) = 0 , ∂yφ(y)− e−iα

4W ′φ(φ(y)) = 0 . (2.92)

Abraham and Townsend [169] studied the intersecting domain wall solutions of (2.92) withW = Φ4 − 4Φ. The generalization of the above equations for the case of supergravity wasstudied in [170]. For W = a2Φ− Φ3

3Eq. (2.92) has kink solution φ = a tanh(ya) [170].

Notice that in the case of special chiral multiplet such a potential would be deformed bythe contribution of an arbitrary imaginary constant, a2 7→ a2 + ic.

When the BPS equations (2.92) are satisfied, the solution of fermionic equations can bewritten in the form [169]

ψ1 = 2χ e−iα/2∂yφ(y) = ψ2 , ψ1 = 2χeiα/2∂yφ(y) = ψ2 (2.93)

with a real Grassmann (fermionic) constant χ,

χ = χ∗ , χχ = 0 . (2.94)

An interesting problem is to study the influence of the supermembrane source on theabove discussed nonsingular domain wall solutions. First observation is that, to maintain thegeneral structure of the solution (2.93) of the fermionic equations, the source contribution inthe r.h.s. of (2.87), σ2

αα∂yψα(y) + i

4ψα(y) W ′′

φφ(φ)(y) = 1

4DαJ(z)|0, should be proportional

to the same real Grassmann constant DαJ(z)|0 ∝ χ. This in its turn suggests the followingansatz for the fermionic coordinates functions

θα(ξ) = uα(ξ)χ , ˆθα(ξ) = uα(ξ)χ (2.95)

with some bosonic functions uα(ξ) = (uα(ξ))∗. Such an ansatz results in that θαθβ =

0 = θα ˆθα and, hence, in that the pull–back of bosonic vielbein simplifies to Ea = dxa.Furthermore, assuming that the normal to the supermembrane worldvolume cannot beorthogonal to the y = x2 axis, we can chose the ‘static gauge’ where x0(ξ) = ξ0 = τ ,x1(ξ) = ξ1, x3 = ξ2 so that the only nontrivial bosonic coordinate function (supermembraneGoldstone field) is identified with x2 = y(ξ). In this gauge

E0 = dξ0 , E1 = dξ1 , E2 = dy(ξ) = dy(ξ0, ξ1, ξ2) , E3 = dξ2 . (2.96)

Furthermore, with such an ansatz J |0 = 0 and all the (quite complicated) components ofcurrent superfield (see appendix B) simplify drastically reducing to their leading terms. Then

29

2.5. Simplest equations of motion for spacetime fields interacting with dynamicalsupermembrane

the problem of finding (particular) solutions of the system of interacting equations looksmanageable.

30

CHAPTER 3

SUPERMEMBRANE INTERACTION WITHDYNAMICAL D=4 N=1 SUPERGRAVITY.

SUPERFIELD LAGRANGIANDESCRIPTION AND SPACETIME

EQUATIONS OF MOTION

In this chapter we study the interacting system of supermembrane and D = 4 N = 1dynamical supergravity. We obtain the complete set of equations of motion for thissystem by varying its complete superfield action. These include the supermembrane

equations, which are formally the same as in the case of supermembrane in supergravitybackground, and the superfield supergravity equations with supermembrane contributions.The existence of a three form potential C3 allowing for a Wess–Zumino term in thesupermembrane part of the action imposes a restriction on the prepotential structure ofminimal supergravity, making its chiral compensator being constructed from a real ratherthan complex prepotential. We will develop the Wess–Zumino type approach for thisspecial minimal supergravity and present its basic variations which are characterized, besidesδHa, by one real variation (δV ) instead a complex ones (δU , δU). The most importantconsecuence of this modification in the prepotential structure is that the right hand side ofthe Einstein equation acquires a term proportional to an integration constant, a dynamicallygenerated cosmological constant. We also present the supergravity superfield equationswith the supermembrane contributions obtained by varying the action with respect to thesuperfields of special minimal supergravity and write these resulting superfield equationsin the special ”WZθ=0” gauge, which is the standard Wess–Zumino gauge completed by

the condition that the supermembrane Goldstone field is set to zero (θ = 0). We solve theauxiliary field equations and show that these result in the effect of dynamical generation ofcosmological constant and its ’renormalization’(due to supermembrane contributions) in

31

3.1. Superfield supergravity and supermembrane in curved D=4, N=1 superspace

such a way that the cosmological constant values in the branches of spacetime separated bythe supermembrane worldvolume are generically different.

3.1. Superfield supergravity and supermembrane incurved D=4, N=1 superspace

3.1.1. Superfield supergravity action, superspace constraints andequations of motion

The superfield action of the minimal off-shell formulation of D = 4, N = 1 supergravity[158]

SSG =

∫d8Z E :=

∫d4xd4θ sdet(EA

M) , (3.1)

is given by the superdeterminant (or Berezinian) of the matrix of supervielbein coefficients,EAM(Z) in (1.13), which obey the set of supergravity constraints. These can be collected

together with their consequences in the following expressions for the superspace torsion2-forms (1.14), (1.15) (see [72] and refs. therein)

T a = −2iσaααEα ∧ Eα − 1

8Eb ∧ EcεabcdG

d , (3.2)

Tα := (T α)∗ =i

8Ec ∧ Eβ(σcσd)β

αGd − i

8Ec ∧ EβεαβσcββR +

1

2Ec ∧ Eb Tbc

α .

(3.3)

The main superfields, real vector Ga = (Ga)∗ and complex scalar R = (R)∗, entering (3.2)

and (3.3), obey

DαR = 0 , DαR = 0 , (3.4)

DαGαα = −DαR , DαGαα = −DαR . (3.5)

These relations can be obtained by studying the Bianchi identities (1.17), which also allowto find the expression for superfield generalization of the gravitino field strength, Tbc

α(Z),

Tαα ββ γ ≡ σaαασbββεγδTab

δ = −18εαβD(α|Gγ|β) − 1

8εαβ[Wαβγ − 2εγ(αDβ)R] , (3.6)

involving one more main superfield, Wαβγ = W(αβγ) =: (Wαβγ)∗. This obeys

DαWαβγ = 0 , DαW αβγ = 0 , (3.7)

DγWαβγ = DγD(αGβ)γ . (3.8)

Studying the Bianchi identities with the constraints (3.2), (3.3) one also finds that thesuperfield generalization of the left hand side of the supergravity Rarita–Schwinger equation

32

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

reads

εabcdTbcασdαα =

i

8σaββD(β|Gβ|α) +

3i

8σaβαDβR , (3.9)

and the superfield generalization of the Ricci tensor is

Rbcac = 1

32(DβD(α|Gα|β) − DβD(βGα)α)σaαασbββ − 3

64(DDR +DDR− 4RR)δab .(3.10)

This suggests that superfield supergravity equation should have the form1

Ga = 0 , (3.11)

R = 0 , R = 0 . (3.12)

Eqs. (3.11) and (3.12) can be obtained by varying the action (3.1) with respect tosupervielbein obeying the supergravity constraints (3.2), (3.3) [158]. Such admissiblevariations are expressed through a vector parameter δHa and complex scalar parameterδU = (δU)∗ which enter the variation of the supervielbein and spin connection under thesymbol of the chiral projector (DαDα − R) (see [72, 84] for more detail). They correspondto the variations of the so-called prepotentials, unconstrained superfields which appear inthe general solution of the supergravity constraints. The minimal supergravity constraintsare solved in terms of the axial vector superfield Hµ [151] and chiral compensator Φ [159].This latter obeys DαΦ = 0 and, hence, can be expressed as Φ = (DαDα − R)U with acomplex unconstrained superfield U .

Thus the set of minimal supergravity prepotentials includes Hµ, U and U = (U)∗ whichare in one to one correspondence with the set of three independent variations δHa, δU andδU = (δU)∗ of the Wess–Zumino approach to supergravity [82, 158] producing the threesuperfield equations (3.11) and (3.12). In short, as it had been known already from [158],

δSSG =

∫d8ZE

[1

6Ga δH

a − 2R δU − 2R δU]. (3.13)

3.1.2. Supermembrane action in minimal supergravity background

As it is well known, the supermembrane action [61, 76](cf. chapter 2) is given by the sumof the Dirac–Nambu–Goto and the Wess–Zumino term,

Sp=2 =1

2

∫d3ξ√g −

∫W 3

C3 = −1

6

∫W 3

∗Ea ∧ Ea −∫W 3

C3 . (3.14)

The former is given by the volume of W 3 defined as integral of the determinant of theinduced metric, g = det(gmn),

gmn = EamηabE

bn , Ea

m := ∂mZM(ξ)Ea

M(Z) . (3.15)

1See [72, 84] for more detail on the superfield description of minimal supergravity in the present notation.

33

3.1. Superfield supergravity and supermembrane in curved D=4, N=1 superspace

Here ξm = (τ, σ1, σ2) are local coordinates on W 3 and ZM(ξ) are coordinates functionswhich determine the embedding of W 3 as a surface in target superspace Σ(4|4),

W 3 ⊂ Σ(4|4) : ZM = ZM(ξ) = (xµ(ξ) , θα(ξ)) . (3.16)

In the second equality of (3.14) the Dirac–Nambu–Goto term is written as an integral ofthe wedge product of the pull–back of the Σ(4|4) bosonic supervielbein form Ea to W 3,

Ea = dξmEam = dZM(ξ)Ea

M(Z) , (3.17)

and of its Hodge dual two form defined with the use of the induced metric (3.15) and itsinverse gmn,

∗ Ea :=1

2dξm ∧ dξn√gεmnkgklEa

l . (3.18)

The second, Wess–Zumino term of the supermembrane action (3.14) describes thesupermembrane coupling to a 3–form gauge potential C3 defined on Σ(4|4),

C3 =1

3!dZM ∧ dZN ∧ dZKCKNM(Z) =

1

3!EC ∧ EB ∧ EACABC(Z) . (3.19)

Thus, to write a supermembrane action, one has to construct the 3–form gauge potential C3

in the target superspace Σ(4|4) and take its pull–back to the supermembrane worldvolume

C3 =1

3!dZM ∧ dZN ∧ dZKCKNM(Z(ξ)) =

1

3!EC ∧ EB ∧ EACABC(Z) =

=1

3!dξm ∧ dξn ∧ dξkCknm = d3ξεmnkCknm . (3.20)

Actually, to study supermembrane in supergravity background, it is sufficient to knowthe field strength of the above 3–form potential, H4 = dC3. This should be closed,dH4 = 0, and supersymmetric invariant 4–form. In flat superspace such a form existsand represents a nontrivial Chevalley–Eilenberg cohomology of the N = 1 supersymmetryalgebra [162, 163, 171](cf. chapter 2).

The minimal supergravity superspace allows for existence of two closed 4-forms

H4L = − i4Eb ∧ Ea ∧ Eα ∧ Eβσab αβ − 1

128Ed ∧ Ec ∧ Eb ∧ EaεabcdR , dH4L = 0 ,(3.21)

and its complex conjugate H4R = (H4L)∗ (see [84]). Its real part,

H4 := dC3 =1

4!EA4 ∧ ... ∧ EA1HA1...A4(Z) = H4L +H4R , (3.22)

is also closed and provides the 4–form field strength associated to the Wess–Zumino (WZ)term of the supermembrane action in the minimal supergravity background [87],

∫W 3 C3 in

(3.14). Indeed, the WZ term can also be defined as an integral of the closed 4 form H4,related to C3 by H4 = dC3, over some four dimensional space W 4 the boundary of which is

34

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

given by the supermembrane worldvolume W 3,∫W 3

C3 =

∫W 4 : ∂W 4=W 3

H4 . (3.23)

The condition that the form H4 is closed, dH4 = 0, guaranties that the integral∫W 4 H4 is

independent on the choice of W 4 and, thus, is related to the supermembrane worldvolumeW 3.

The fact that the knowledge of H4 is completely sufficient for studying the propertiesof closed supermembrane in a supergravity background is related to that in this case theonly dynamical variables are the supermembrane coordinate functions ZM(ξ), that theaction is written in term of pull–back of differential forms to W 3, and that the variationof the differential form with respect to the coordinates can be calculated with the useof the Lie derivative formula, in particular δδZC3 = iδZH4 + diδZC3 = 1

3!EA4 ∧ ... ∧

EA2 δZM EA1M HA1...A4(Z) + d(1/2EC ∧ EB δZM EA

M CABC(Z)). 2

The supermembrane equations of motion in the minimal supergravity background, whichare obtained by varying the action (3.14) with respect to the coordinate functions δZM(ξ),

δSp=2 =

∫W 3

(1

2M3aEM

a(Z) + iΨ3αEMα(Z) + iΨ3αEM

α(Z)

)δZM(ξ) , (3.24)

read

M3 a := D ∗ Ea + iEb ∧ Eα ∧ Eβσabβα − iEb ∧ ˆEα ∧ ˆEβσabβα −

−1

8Eb ∧ Ec ∧ Edεabcd(R + R) = 0 (3.25)

and

Ψ3α := ∗Ea ∧(Eασaαα − (˜γσa)αβ

ˆEβ)

= 0 , (3.26)

Ψ3α := ∗Ea ∧(σaαα

ˆEα + Eβ(σa ˜γ)αβ

)= 0 , (3.27)

where the matrix γβα is defined by

γβα = εβαεαβ ˜γβα =i

3!√gσaβαεabcdε

mnkEbmE

cnE

dk = −(γαβ)∗ (3.28)

and obeys

γβα ˜γαα = δβα , ˜γααγαβ = δαβ . (3.29)

2For closed supermembrane ∂W 3 = ∅ so that∫W 3 dα2 =

∫∂W 3 α2 = 0 for any 2-form α2, including for

α2 = iδZC3.

35

3.1. Superfield supergravity and supermembrane in curved D=4, N=1 superspace

Some identities involving the above matrix are

γσa = −σa ˜γ +i

3!√gεabcdε

mnkEbmE

cnE

dk , (3.30)

∗Eaγσaγ = ∗Eaσa , ∗Eaγσa = − ∗ Eaσa ˜γ , (3.31)1

2Eb ∧ Ea ∧ Eβ σabβα = ∗Ea ∧ Eβ(σa ˜γ)βα , (3.32)

1

2Eb ∧ Ea ∧ ˆEβσabβα = − ∗ Ea ∧ ˆEβ(σaγ)βα . (3.33)

They are useful, in particular, to show that the fermionic equations of motion obey theNoether identity Ψ3α = Ψα

3 γαα reflecting the local fermionic κ–symmetry

δκZM = κα(ξ)(EM

α (Z) + γααεαβEM

β(Z)) (3.34)

with the local fermionic “parameter” κα(ξ) = (κα)∗ obeying

κα(ξ) = −κα(ξ)˜γαα ⇔ κα(ξ) = −κα(ξ)γαα . (3.35)

The relation of the supermembrane κ–symmetry in curved superspace with the minimalsupergravity constraints was discussed in [87]. The flat superspace limit of our equationsreproduces the equations of the seminal paper [76].

3.1.3. 3–form potential in the minimal supergravity superspace.Special minimal supergravity

Thus, as we have seen in the previous subsection, to find the equations of motion ofsupermembrane in supergravity background as well as to study its symmetries it is sufficientto know the closed 4-form H4 = dC3 in the background superspace.

However, to calculate the supermembrane current(s) describing the supermembranecontribution(s) to the supergravity (super)field equations, one needs to vary the Wess–Zumino term

∫C3 of the supermembrane action with respect to the supergravity (super)fields.

Thus one arrives at a separate problem of finding the variation

δC3 =1

3!EC ∧ EB ∧ EAβABC(δ) (3.36)

such that dδC3 = δH4 reproduces the variation of H4 from (3.22), (3.21), written in termsof the basic supergravity variations (we refer to Appendix C for the explicit expression ofδH4).

Studying such a technical problem we have found that it imposes a restriction on theindependent variations of the supergravity prepotentials, or equivalently, on the independentparameters of the admissible supervielbein variations, thus transforming the generic minimalsupergravity into a special minimal supergravity. This off–shell supergravity formulation hadbeen described for the first time in [85], further discussed in [86] (see also latter [172]) andelaborated in [87] using the elegant combination of superfield results and the component

36

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

’tensor calculus’ approach on the line of [155].

In [84] we described this special minimal supergravity in the complete Wess–Zuminosuperfield formalism. Referring to that paper for technical details, we only notice thatthe existence of the 3-form potential imposes a restriction on the prepotential structureof minimal supergravity which in our approach manifests itself in that the basic complexvariations δU and δU = (δU)∗ are expressed in terms of one real variation δV , essentially

δU =i

12δV , δU = − i

12δV . (3.37)

As a result, the variation of the special minimal supergravity action is essentially (seeAppendix C)

δSSG =1

6

∫d8ZE

[Ga δH

a + (R− R)iδV]. (3.38)

Hence the set of superfield equations of special minimal supergravity still includes the vectorsuperfield equation (3.11),

Ga = 0 , (3.39)

but instead of the complex scalar superfield equations (3.12), valid in the case of genericminimal supergravity, in the case of special minimal supergravity we have only the real scalarequation

R− R = 0 . (3.40)

Clearly, due to chirality of R, DαR = 0, and anti-chirality of R, DαR = 0, the above Eq.(3.40) also implies that d(R+ R) = 0 so that on the mass shell the complex superfield R isactually equal to a real constant,

R = 4c , R = 4c , c = const = c∗ . (3.41)

Using (3.10), one finds that the superfield equation (3.40) results in Einstein equation withcosmological constant

Rbcac = 3c2δb

a . (3.42)

The value of the cosmological constant is proportional to the square of the above arbitraryconstant c, which has appeared as an integration constant, so that the special minimalsupergravity is characterized by a cosmological constant generated dynamically.

The above mechanism of the dynamical generation of cosmological constant in specialminimal supergravity is the same as was observed by Ogievetski and Sokatchev [89] intheir theory of axial vector superfield. In the language of component spacetime approachto supergravity the dynamical generation of cosmological constant in the special minimalsupergravity was described in [87] and before it, in purely bosonic perspective in [88, 91, 92]and in the context of spontaneously broken N = 8 supergravity in [90].

37

3.2. Superspace action and superfield equations of motion for the interacting system ofdynamical supergravity and supermembrane

3.2. Superspace action and superfield equations ofmotion for the interacting system of dynamicalsupergravity and supermembrane

The action for interacting system of dynamical supergravity and supermembrane reads

S = SSG + T2Sp=2 =

∫d8ZE(Z) +

T2

2

∫d3ξ√g − T2

∫W 3

C3 , (3.43)

where Sp=2 is the same as in Eq. (3.14), the supervielbein (1.13) and the 3-form potential(3.20) are assumed to be restricted by the minimal supergravity constraints (3.2), (3.3),(3.22), (3.21). Furthermore, as we have discussed in previous sections (and in more detailsin Appendix C), the existence of the 3-form potential imposes the restrictions (3.37) on theprepotentials of minimal supergravity or equivalently on the basic supergravity variations.

As a result, the superfield equations which appear as a result of variation of the interactingaction (3.43) read

Ga = T2Ja , (3.44)

and

R− R = −iT2X (3.45)

where Ja and X = (X )∗ are supermembrane scalar superfields. Roughly speaking, they areobtained as a result of varying the supermembrane action with respect to the prepotentialsof the special minimal supergravity, this is to say as δSp=2/δH

a and δSp=2/δV , and havethe form

Ja =

∫W 3

3

EEb ∧ Eα ∧ Eβ σabαβδ

8(Z − Z)−

−∫W 3

3i

E

(∗Ea ∧ Eα +

i

2Eb ∧ Ec ∧ ˆEβεabcdσ

dβα

)Dαδ8(Z − Z) + c.c−

−∫W 3

i

8EEb ∧ Ec ∧ Ed εabcd

(DD − 1

2R

)δ8(Z − Z) + c.c.+

+

∫W 3

1

4E∗ Eb ∧ EbGa δ

8(Z − Z)−

−∫W 3

1

4E∗ Ec ∧ Ebσdαα

(3δcaδ

db − δdaδcb

)[Dα, Dα]δ8(Z − Z) , (3.46)

38

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

and

X =6i

E

∫W 3

Ea ∧ Eα ∧ ˆEα σaαα δ8(Z − Z)−

−3

2

∫W 3

Eb ∧ Ea ∧ Eα

Eσabα

βDβδ8(Z − Z) + c.c+

+

∫W 3

Eb ∧ Ec ∧ Ed

8Eεabcdσ

aαα[Dα, Dα]δ8(Z − Z) +

+i

∫W 3

∗Ea ∧ Ea

4E

(DD − R

)δ8(Z − Z) + c.c.+

+

∫W 3

1

4EEb ∧ Ec ∧ EdεabcdG

a δ8(Z − Z) . (3.47)

Notice that, as a consequence of (3.5), the supermembrane current superfields obey

DαJαα = iDαX , DαJαα = −iDαX . (3.48)

Although at first glance these relations look different from any of listed in [173, 174], theycan be reduced to the Ferrara–Zumino multiplet [175] if one takes into account Eq. (3.45).Indeed, this states that the real superfield X in the r.h.s. of Eq. (3.48) is the sum of chiralsuperfield (equal to iR) and its complex conjugate, so that only the first (second) onecontributes to the r.h.s. of the first (second) equation in (3.48).

3.3. Spacetime component equations of the D = 4

N = 1 supergravity–supermembrane interactingsystem

3.3.1. Wess–Zumino gauge plus partial gauge fixing of the localspacetime supersymmetry (WZθ=0 gauge)

The structure of the current superfields (3.46), (3.47) is quite complicated. So is thestructure of their components. To simplify the supercurrent components which contributeto the equations of physical, spacetime component fields, we use the superspace general

39

3.3. Spacetime component equations of the D = 4 N = 1 supergravity–supermembraneinteracting system

coordinate invariance to fix the Wess–Zumino (WZ) gauge on supergravity superfields,

iθEα := θαEα

α = θα , iθEα := θαEα

α = θα , (3.49)

θα := θβδ αβ, θα := θβδ α

β, (3.50)

iθEa := θαEα

a = 0 , (3.51)

iθwab := θβwab

β= 0 (3.52)

(see [72] for references and more detail) and the (pull–back to W 3 of the) local spacetimesupersymmetry to set to zero the fermionic Goldstone field of the supermembrane,

θα(ξ) = 0 ⇔ θα(ξ) = 0 , ˆθα(ξ) = 0 . (3.53)

A detailed discussion on this ”WZθ=0” gauge can be found in [71–74]. We notice only fewof its properties.

Firstly, in the WZ gauge (3.49), (3.51) the leading component of supervielbein matrixhas a triangular form,

ENA|θ=0 =

(eaν(x) ψαν (x)

0 δβα

)⇒ EA

N |θ=0 =

(eνa(x) −ψβa (x)

0 δαβ

), (3.54)

which implies, in particular, the following relation between the leading component of Tabα

and the true gravitino field strength D[µψαν](x)

Tabα|θ=0 = 2eµae

νbD[µψ

αν](x)− i

4(ψ[aσb])βG

αβ|θ=0 − i4(ψ[aσb])

αR|θ=0 . (3.55)

Secondly, we would like to comment on symmetries leaving Eqs. (3.49)–(3.53) invariant.The WZ gauge (3.49), (3.51), (3.52) is preserved by spacetime diffeomorphisms, localLorentz symmetry and supersymmetry. Fixing further the gauge (3.53) we break 1/2 of thelocal supersymmetry on the worldvolume of the supermembrane. The only restriction on theparameter of the local spacetime supersymmetry εα(x) is the condition that its pull–back toW 3, εα := εα(x(ξ)), and its complex conjugate ˆεα := εα(x(ξ)) are related by

εα = ˆεα ˜γαα , (3.56)

where ˜γαα is the supermembrane κ–symmetry projector (3.28) calculated with θ(ξ) = 0. Eq.(3.56) is tantamount to saying that the pull–back of the local supersymmetry parameter toW 3 is expressed through the κ–symmetry parameter of the supermembrane. There are norestrictions on the local supersymmetry parameter outside the supermembrane worldvolumeso that the equations (3.56) can be understood as the boundary condition imposed on thesupersymmetry parameter on the domain wall W 3.

40

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

3.3.2. Current superfields in the WZθ=0 gauge. Currentprepotentials and Rarita–Schwinger equation

In the gauge (3.49)–(3.56),

Ea = ea = dxµeµa(x) , Eα = ψα = dxµψµ

α(x) , (3.57)

and

Dαδ8(Z − Z) =1

8θα θθ δ

4(x− x)+ ∝ θ∧4 ,

Dαδ8(Z − Z) = −1

8θα θθ δ

4(x− x) + θ∧4 , (3.58)

DαDαδ8(Z − Z) = −1

4θθ δ4(x− x)+ ∝ θ θθ ,

DαDαδ8(Z − Z) = −1

4θθ δ4(x− x)+ ∝ θθ θ , (3.59)

[Dα, Dα]δ8(Z − Z) = −1

2θα θα δ

4(x− x)+ ∝ θ∧3 , (3.60)

where θ∧4 := θθ θθ and θ∧3 denotes terms proportional to either θθ θ or θ θθ (or both, whichimplies ∝ θθ θθ). Using these relations and introducing the current pre-potential fields

Pab(x) :=

∫W 3

1

e∗ ea ∧ eb δ4(x− x) , (3.61)

Pa(x) :=

∫W 3

1

eεabcde

b ∧ ec ∧ ed δ4(x− x) =

= eµa(x)

∫W 3

εµνρσdxν ∧ dxρ ∧ dxσ δ4(x− x) , (3.62)

we find that the vector and scalar current superfields (3.46), (3.47) have the form

Jαα|θ=0 =θβ θβ

8( 3Pab(x)σaαασ

ββb − 2δα

βδαβPbb(x))− i(θθ − θθ)

32σaααPa(x)+ ∝ θ∧3 (3.63)

and

X|θ=0 = −θσaθ

16Pa + i

(θθ − θθ)16

Paa(x)+ ∝ θ∧3 . (3.64)

Using (3.63) and (3.64) one can easily check that Eqs. (3.48) are satisfied at lowest orderin θ.

One also sees that there is no explicit supermembrane contributions to the Rarita–Schwinger equations of the supergravity–supermembrane interacting system which thus

41

3.3. Spacetime component equations of the D = 4 N = 1 supergravity–supermembraneinteracting system

reads

εµνρσeaν(x)Dρψασ (x)σaαα = 0 . (3.65)

However, such a contribution is actually present in (3.65) implicitly, hidden inside thecovariant derivative. Indeed, as indicated by Einstein equation, the bosonic vielbein and thespin connection do contain some contributions from supermembrane.

3.3.3. Einstein equation of the supergravity–supermembraneinteracting system in the WZ θ=0 gauge

The Einstein equation with supermembrane current contributions can be obtained as leadingterm in the decomposition of Eq. (3.10), i.e.

Rbcac|

θ=0=

1

32(DβD(α|Jα|β) − DβD(βJα)α)|

θ=0σaαασbββ −

3i

64(DDX − DDX )|

θ=0δab +

+3

16(RR)|

θ=0δab . (3.66)

The first two terms in the r.h.s. of Eq. (3.66) can be easily calculated from Eqs. (3.63),(3.64), while the last term, in the light of that the scalar superfield equation has the formof Eq. (3.45), is expressed in terms of (R + R)2|

θ=0and requires a separate study. As an

intermediate resume let us fix that

Rbcac|

θ=0 , θ=0= − 3

32T2

(Pba(x)− 1

2δabPcc(x)

)+

3

64(R + R)2|

θ=0δab . (3.67)

The last term is the square of (R + R)|θ=0

which, as a result of (3.45), obeys the equation

∂µ(R + R)|θ=0

=T2

16

∫W 3

εµνρσdxν ∧ dxρ ∧ dxσ δ4(x− x) . (3.68)

The solution of this equation can be written in the form

R(x) + R(x) = 8c+T2

16

x∫x0

dxµ∫W 3

εµνρσdxν ∧ dxρ ∧ dxσ δ4(x− x) , (3.69)

where c is an arbitrary constant which corresponds to the value of (R+ R) at the spacetimepoint xµ0 providing the lower limit of the integral in the second term, c = (R(x0)+ R(x0))/8.

One can easily check that

Θ(x, x0|x) :=

x∫x0

dxµ∫W 3

εµνρσdxν ∧ dxρ ∧ dxσ δ4(x− x) , (3.70)

42

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

entering the second term in the r.h.s. of (3.69), obeys

∂µΘ(x, x0|x) =

∫W 3

εµνρσdxν ∧ dxρ ∧ dxσ δ4(x− x) . (3.71)

Furthermore, using a convenient local frame in the neighborhood of the worldvolume, onecan check that Θ(x, x0|x) vanishes if the points xµ and xµ0 are on the same side of spacetimewith respect to the domain wall provided by the supermembrane worldvolume while it isequal to ±1 if these points belongs to the different branches of the spacetime separated bythis domain wall. This is to say that Eq. (3.70) defines a counterpart of the Heaviside stepfunction associated to the direction orthogonal to the supermembrane worldvolume. The laststatement about association implies that Θ(x, x0|x) is a functional of the supermembranecoordinate function xµ(ξ). Furthermore, as in the case of the standard Heaviside stepfunction Θ(y), we can use 3 (Θ(x, x0|x))2 = Θ(x, x0|x).

Thus our solution of the auxiliary field equations (3.69) can be written as (cf. [88])

R(x) + R(x) = 8c+T2

16Θ(x, x0|x) (3.72)

and implies that Eq. (3.67) reads

Rbcac(x) = −3T2

32

(Pba(x)− 1

2δabPcc(x)

)+

+3δab

(c2 +

((T2

128+ c

)2

− c2

)Θ(x, x0|x)

), (3.73)

where Pba(x) is the singular contribution defined in (3.62).

3.3.4. Cosmological constant generation in the interacting systemand its “renormalization” due to supermembrane

Let us analyze the supermembrane contribution to the Einstein equations. These can beseparated in two classes, one containing singular contributions and the other containingregular contributions proportional to T2.

Being a bit more provocative one can say about three classes, counting also the contributionproportional to square of the arbitrary integration constant c, as far as this comes from theauxiliary field sector of the special minimal supergravity, the off–shell formulation whichis ’elected’ by the supermembrane. As we have already commented in sec. 3.1.3, thesupermembrane can exist in a background of a generic minimal supergravity, however the

3In the case of standard standard Heaviside step function this is equivalent to setting the indefinite valueΘ(0) equal to 1/2. Indeed, calculating the derivative ∂y(Θ(y)Θ(y)) = 2Θ(y)δ(y) = 2Θ(0)δ(y) we findthat this coincides with ∂yΘ(y) = δ(y) when Θ(0) = 1/2. In our case Θ(x, x0|x) is the counterpart ofeither +Θ(y) or −Θ(y) so that (Θ(x, x0|x))2 = ±Θ(x, x0|x). However, by a suitable choice of thelocation of the point x0 with respect to W 3 one can always arrive at the situation with nonnegativeΘ(x, x0|x). Below for simplicity we assume this choice is made.

43

3.3. Spacetime component equations of the D = 4 N = 1 supergravity–supermembraneinteracting system

supermembrane interaction with dynamical supergravity requires this to be special minimalsupergravity. This in its turn, even in the absence of any matter (neither of the fieldtheoretical type nor of branes), produces Einstein equations with a cosmological constantgenerated dynamically. Then this cosmological constant proportional to the square of theabove arbitrary integration constant c should also be considered as a(n indirect) contributionof the supermembrane to the Einstein equation.

To be more concrete, Eq. (3.73) can be written in the form Rbcac(x) = 3c2δab+ ∝ T2,

Racbc(x) = ηab 3c2 + T2

(T singab (x) + T regab (x)

). (3.74)

When T2 is set to zero, it contains a nonvanishing cosmological constant contribution withΛ = −3c2. This (AdS-type) cosmological constant is generated dynamically as far as it isproportional to the (minus) square of the arbitrary integration constant c which is inevitablein the special minimal supergravity equations due to its auxiliary field structure (see [87]and [84] for references and more discussion). In its turn, special minimal supergravity,and not generic minimal supergravity can be included into the action of the supergravity–supermembrane interacting system. In this sense the cosmological constant generateddynamically is the first ’relict’ contribution from the supermembrane to the Einstein equationof the interacting system.

The second type of the supermembrane contributions to the r.h.s. of the Einstein equationare singular terms ∝ Pcd(x) (3.62),

T singab (x) = −T23

32

(Pba(x)− 1

2ηbaPcc(x)

)=

= −3T2

32

∫W 3

1

e∗ ea ∧ eb δ4(x− x) +

3T2

64ηba

∫W 3

1

e∗ ec ∧ ec δ4(x− x) (3.75)

which are expected when (super)gravity interacts with supermembrane.In the third type we collect the regular supermembrane contributions which are proportional

to the supermembrane tension,

T regab (x) = ηabT reg(x) , T reg(x) = +3T2

64

(T2

256+ c

)Θ(x, x0|x) . (3.76)

To appreciate the role of this contribution it is instructive to consider the Einstein equationin two pieces of the spacetime separated by the supermembrane worldvolume. Let us denotethe half-space where Θ(x, x0|x) = 1 by M4

+ and the half-space where Θ(x, x0|x) = 0 byM4−. Then the singular terms (3.75) do not contribute and the Einstein equation reads

M4+ : Racb

c(x) = 3ηab

(T2

128+ c

)2

. (3.77)

M4− : Racb

c(x) = 3ηab c2 . (3.78)

An evident observation is that, in the general case, the cosmological constants in differentbranches of spacetime separated by the worldvolume W 3 are different.

44

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

One also notices that the cosmological constants in M4+ and M4

− coincide if c = − T2256

.However, as far as c is an arbitrary integration constant, fixing its value is equivalent toimposing a kind of boundary conditions and we do not see any special reason to chose suchboundary conditions in such a way that c = − T2

256. Rather we should allow a generic value

of c and thus accept that the cosmological constant takes different values in the branchesof spacetime separated by the supermembrane worldvolume.

Notice that the solution of the Einstein equation describing membranes separating twoAdS5 spaces with different values of cosmological constants were studied in [93], as a Braneworld alternative to the dark energy, and [94] in relation with the hypothesis on possiblechange of signature in the Brane World models. See also [95] for the related studies. In thebosonic perspective the appearance of different cosmological constants on the different sidesof a domain wall interacting with gravity and a 3–form gauge field was known from [88],where it was used as a basis for a bag model for hadrons, and from [92] where this effectwas proposed as a mechanism for damping the cosmological constant. Our present studyindicates that the result on the different values of cosmological constant on the differentsides of the supermembrane domain wall is an imminent consequence of the dynamics ofthe supersymmetric interacting system of the supermembrane and dynamical D = 4 N = 1supergravity.

3.4. On supersymmetric solutions of the interactingsystem equations

When searching for purely bosonic supersymmetric solutions, setting ψαµ = 0, one studiesthe Killing spinor equations, which appears as the conditions of supersymmetry preservation,δεψ

αµ = 0. When starting from superfield formulation of supergravity, δεψ

αµ can be calculated

with the use of superspace Lie derivative, this is to say δεψαµ = Dµε

α + (ECµ ε

βTβCα)|θ=0.

Hence, in a generic off-shell D = 4 N = 1 minimal supergravity background the Killingequations are

Dεα +i

8ec(εσcσd)β

α Gd|θ=0 +i

8ec (εσc)

α R|θ=0 = 0 (3.79)

and the complex conjugate equation. Using the superfield equations of motion (3.44),(3.45), the explicit form of the current superfields in the WZθ=0 gauge, Eqs. (3.63), (3.64),and Eq. (3.72), we find that the Killing equation (3.79) reads

Dεα +i

2ea (εσa)

α

(c+

T2

128Θ(x, x0|x)

)= 0 . (3.80)

45

3.4. On supersymmetric solutions of the interacting system equations

We can split this on two Killing equations valid in two different branches of spacetimeseparated by the supermembrane worldvolume,

M4− : Dεα +

i

2ea (εσa)

α c = 0 , (3.81)

M4+ : Dεα +

i

2ea (εσa)

α

(c+

T2

128

)= 0 . (3.82)

The supersymmetry parameter should also obey the boundary conditions (3.56) on theworldvolume W 3, which is the common boundary of M4

+ and M4−,

W 3 = ±∂M4± : εα = ˆεα ˜γαα , εα := εα(x(ξ)) , ˆεα := εα(x(ξ)) . (3.83)

The detailed study of these system of Killing spinor equations and the search for thesupersymmetric solutions of the interacting system equations on their basis is an interestingsubject for future. An intriguing question is whether the supersymmetric solutions of theequations of the interacting system exist in the generic case of arbitrary c correspondingto different values of cosmological constants on different sides of the supermembraneworldvolume, or supersymmetry selects some particular values of the constant c. Presentlywe can state that if obstructions existed, they would occur due to the singular terms withsupport on the worldvolume W 3, while the mere fact of different values of cosmologicalconstant on the branches of spacetime situated on the different sides of W 3 does notprohibit supersymmetry. Indeed, let us study the integrability conditions for the Killingspinor equations in M4

±. Applying the exterior covariant derivatives to Eqs. (3.81) and(3.82) and using the Ricci identities DDεα = −1

4Rabεβσab β

α and the equations complexconjugate to (3.81) and (3.82), we find

M4− : Rabεβσab β

α =1

4|c|2ed ∧ ec εβσcd βα , (3.84)

M4+ : Rabεβσab β

α =1

4

∣∣∣∣c+T2

128

∣∣∣∣2 ed ∧ ec εβσcd βα . (3.85)

If we search for a purely bosonic solution preserving all the supersymmetry in M4− and M4

+,Eqs. (3.84) and (3.85) should be obeyed for an arbitrary εα. This implies

M4− : Rcd

ab =1

2|c|2δ[c

aδd]b , (3.86)

M4+ : Rcd

ab =1

2

∣∣∣∣c+T2

128

∣∣∣∣2 δ[caδd]

b , (3.87)

i.e. that M4± are AdS spaces with apparently different cosmological constants. One can

easily check that (3.86) and (3.87) solve our equations of motion (3.78) and (3.77) andthus describe the completely supersymmetric solution of the system of the supergravityequations of the interacting system (at least) when these are considered modulo singularterms with the support on W 3.

Let us stress that such a system of equations does contain the supermembrane con-

46

Chapter 3. Supermembrane interaction with dynamical D=4 N=1 supergravity. SuperfieldLagrangian description and spacetime equations of motion

tributions: not only an indirect, which comes from an arbitrary cosmological constantgenerated dynamically due to the structure of the supergravity auxiliary fields imposed bythe supergravity interaction with supermembrane (see [87] and also [88–91]), but also direct,which is a shift of cosmological constant on one of the sides of the brane worldvolume onthe value which is proportional to the supermembrane tension (see [88, 92]). Furthermore,although preserving all 4 supersymmetries in M4

− and M4+, when considered as a solution of

the equations of interacting system, Eqs. (3.86) and (3.87) describe the 1/2 BPS state, i.e.the state preserving 1/2 of the supersymmetry. Indeed, when considering the interactingsystem we have to restrict the local supersymmetry parameter by the boundary conditions(3.83) on W 3 and these clearly break 1/2 of the supersymmetry on W 3.

47

CHAPTER 4

CONFORMAL HIGHER SPIN THEORY INEXTENDED TENSORIAL SUPERSPACE

In this chapter we present the superfield equations in tensorial N –extended superspacesto describe the N = 2, 4, 8 supersymmetric generalizations of free conformal higherspin theories. We obtain them by quantizing a superparticle model in N–extended

tensorial superspace. We show that no nontrivial generalizations of Maxwell and Einsteinequations to tensorial space appear because N–extended higher spin supermultiplets justcontain additional scalar and spinor fields which obey the standard higher spin equationsin their tensorial space version. We find also that these additional fields appear in thebasic superfield under derivatives so that the theory is invariant under Peccei–Quinn–likesymmetries.

4.1. Superparticle in N -extended tensorial superspace

Let us begin by presenting a simple dynamical model in extended tensorial superspace thequantization of which produces the supersymmetric higher spin field equations.

4.1.1. An action for the Σ(n(n+1)2 |N n) superparticle

The action for a superparticle in extended tensorial superspace Σ(n(n+1)

2|N n) has the form

S =

∫dτ L =

∫dτ [

˙Xαβ(τ)− i ˙

θαI(τ)θβI(τ)]λα(τ)λβ(τ) ,

α, β = 1, ..., n ,

I = 1, 2, ...,N ,(4.1)

49

4.1. Superparticle in N -extended tensorial superspace

where the λα(τ) are auxiliary commuting spinor variables, Xαβ(τ) = Xβα(τ) and θαI(τ)are the bosonic and fermionic coordinate functions which define the superparticle worldline

W 1 ∈ Σ(n(n+1)

2|N n) : ZM = ZM(τ) = (Xαβ(τ) , θαI(τ)), (4.2)

and the dot denotes derivative with respect to proper time τ .It is convenient to write the action (4.1) in the form

S =

∫W 1

Παβλα(τ)λβ(τ) , (4.3)

where

Παβ(τ) = dτ Παβτ (τ) = dτ(

˙Xαβ − i ˙

θI(αθβ)I) ,

α, β = 1, ..., n ,

I = 1, 2, ...,N ,(4.4)

is the pull-back to W 1 of the vielbein Παβ of the flat N -extended tensorial superspace

Σ(n(n+1)

2|Nn)

Παβ = dXαβ − idθI(αθβ)I . (4.5)

The superparticle action is manifestly invariant under rigid supersymmetry of the N -

extended tensorial superspace Σ(n(n+1)

2|Nn) ,

δεXαβ = iθI(αεβ)I , δεθ

Iα = εβI , (4.6)

which acts on the worldline fields as

δεXαβ = iθI(αεβ)I , δεθ

Iα = εβI ; δελα = 0 . (4.7)

The action (4.1) is also manifestly invariant under the GL(n,R) transformations of theα, β = 1, ..., n indices, which reduce to the n-dimensional representation of Spin(1, D − 1)when these indices are thought of as Lorentz-spinorial ones1.

4.1.2. Symplectic supertwistor form of the action

Actually, the Σ(n(n+1)

2|Nn) superparticle action (4.1) is invariant under the larger OSp(N|2n)

supergroup. To make this manifest as well as to determine easily the number of physicaldegrees of freedom it is convenient to use Leibniz’s rule2 to rewrite the action (4.1) in theform

S =

∫W 1

(λαdµα − µαdλα − idχI χI) =

∫W 1

dΥΣΞΣΩΥΩ . (4.8)

1In [98, 99] the counterparts of λα were called ‘s-vectors’ to avoid their immediate identification asGL(n,R) vectors or SO(1, D − 1) spinors.

2It is sufficient to use λαλβdXαβ = λαd(λβX

αβ)− λαXαβdλβ ; no integration by parts is needed.

50

Chapter 4. Conformal higher spin theory

This action is written in terms of the bosonic spinor λα(τ), which is present in (4.1), asecond bosonic spinor µα and N real fermionic variables χI ; these form the N -extendedorthosymplectic supertwistor (see [97, 110] for N = 1)

ΥΣ =(µα λα χI

), α = 1, . . . , n , I = 1, . . . ,N . (4.9)

This generalizes the Penrose twistors [176] (or conformal SU(2, 2) spinors) and the Ferber-Schirafuji supertwistors [177, 178] (carrying the basic representation of D=4 SU(N|2, 2)).The ΥΣ’s carry the defining representation of the OSp(N|2n) supergroup, the transforma-tions of which preserve the (2n+N )× (2n+N ) orthosymplectic ‘metric’ ΞΣΩ,

ΞΣΩ =

0 δαβ 0

−δαβ 0 00 0 −iδIJ

, α = 1, . . . , n , I = 1, . . . ,N . (4.10)

In fact, OSp(1|2n) may be considered as a supersymmetric generalization of the supercon-formal group for D = n

2+ 2 (see [179] and [97, 111, 180] and refs. therein).

The relations between the supertwistor components and the variables of the action (4.1)that make the transition between both actions are

µα = Xαβλβ −i

2θαIχI , χI = θαI λα , (4.11)

which generalize the Penrose and the Ferber-Shirafuji incidence relations [176–178] (see [110]and [97] for N = 1). Since the action (4.8) does not possess any gauge symmetries, thecomponents of the orthosymplectic supertwistors are the true, physical degrees of freedom(2n bosonic and N fermionic) of our generalized superparticle model.

4.1.3. Gauge symmetries of the original action

By construction, the actions (4.8) and (4.1) describe the same dynamical system. Thus,

since the action (4.1) depends on n(n+1)2

+ n bosonic variables and Nn fermionic ones,it should possess n(n− 1)/2 bosonic gauge symmetries and N (n− 1) fermionic ones toreduce the number of degrees of freedom to those of the supertwistors ΥΣ appearing inthe action (4.8). The simplest way to describe these gauge symmetries, called fermionicκ-symmetry and bosonic b-symmetry, is to define restrictions on the basic variations of thebosonic and fermionic coordinate functions (see [97] for the N = 1 superparticle case and[111] for the case of superstring in tensorial superspace.)

δκXαβ = iδκθ

I(αθβ)I , δκθαIλα = 0 , δκλα = 0 , (4.12)

δbXαβλα = 0 , δbθ

Iαλα = 0 , δbλα = 0 . (4.13)

51

4.1. Superparticle in N -extended tensorial superspace

4.1.4. Constraints and their conversion to first class

In the hamiltonian formalism, the δκ and δb gauge symmetries in Eqs. (4.12), (4.13) aregenerated by first class constraints which may be extracted from the following bosonic andfermionic primary constraints of the model (4.1)

dαI := παI + iPαβθβI ≈ 0 , (4.14)

Pαβ := Pαβ − λαλβ ≈ 0 , Pα(λ) ≈ 0 , (4.15)

where

Pαβ :=1

2

δL

δ˙Xαβ

, P (λ)α :=

δLδλα

, παI :=δL

δ˙θαI

, (4.16)

are the canonical momenta conjugated to the coordinate functions and to the auxiliarybosonic spinor (s-vector). Using the canonical Poisson brackets

[Xγδ, Pαβ]PB = −[Pαβ, Xγδ]PB = δα

(γδβδ) , (4.17)

[λβ, Pα(λ)]PB = −[Pα(λ), λβ]PB = δβ

α , (4.18)

παI , θβJPB = θβJ , παIPB = −δαβδIJ , (4.19)

it follows that the nonvanishing Poisson brackets of the above constraints are

dαI , dβJPB = −2iPαβδIJ , [Pαβ, P γ(λ)]PB = −2λ(αδβ)γ . (4.20)

These clearly indicate that the primary constraints above are a mixture of first and secondclass constraints. Rather than separating them, we use below the so-called ‘conversionprocedure’ [97, 181–187] by which a pair of degrees of freedom is added to each pair ofsecond class constraints to modify Eqs. (4.20) in such a way that they form a closed algebra.In this way, these modified constraints become first class ones generating gauge symmetriesin the enlarged phase space. In it, all the constraints of the model are first class and accountas well for the original second class constraints. These can be recovered by gauge fixingthe additional gauge symmetries/first class constraints of the system in the enlarged phasespace. For the N = 1 version of (2.1) this was done in [97].

As the bosonic sector of all the superparticle models is the same irrespective of N , wemay use the results of [97] for N = 1 and state that the conversion in the bosonic sectoris effectively reduced to ignoring the constraints Pα(λ) ≈ 0 in the analysis. An easy wayto see that this is indeed consistent is to observe that, as far as λα 6= (0, ..., 0) (the usualconfiguration space restriction for twistor-like variables), the second brackets in (4.20) showthat Pα(λ) = 0 is a good gauge fixing condition for n of n(n + 1)/2 gauge symmetriesgenerated by the constraints Pαβ.

To perform the conversion in the fermionic sector, we introduce the N fermionic variablesχI and postulate for them the Clifford-type Poisson brackets

χI , χJPB = −2iδIJ . (4.21)

52

Chapter 4. Conformal higher spin theory

These χI are then used to modify the fermionic constraints to DαI = dαI + iχIλα. Thus,after conversion, the superparticle model (4.1) is described by the following set of first classconstraints

DαI := παI + iPαβθβI + iχIλα ≈ 0 , Pαβ := Pαβ − λαλβ ≈ 0 , (4.22)

which obey the superalgebra of the rigid supersymmetry of N -extended tensorial superspace

Σ(n(n+1)

2|Nn),

DαI ,DβJPB = −2iPαβδIJ , [Pαβ,DγI ]PB = 0 , [Pαβ,Pγδ]PB = 0 . (4.23)

4.2. Quantization of the superparticle in Σ(n(n+1)

2 |Nn)

with even N and conformal higher spin equations

Quantizing the model in its orthosymplectic-twistorial formulation (4.8) is straightforward.The canonical hamiltonian is equal to zero and thus the Schrodinger equation simply statesthat the wavefunction is independent of τ . Following a procedure similar to that in [97] onecan show that, in the n = 4 tensorial space corresponding to D = 4, the wavefunction of thebosonic limit of the superparticle model (4.8) describes the solution of the free higher spinequations. This means that it can be written in terms of an infinite tower of left and rightchiral fields φA1...A2s(pµ) and φA1...A2s

(pµ) for all half-integer values of s with pµpµ = 0.

Let N > 1 and even (as it will be henceforth). Quantization a la Dirac of a dynamicalsystem with first class constraints requires imposing them as equations to be satisfied byits wavefunction. In the case of our superparticle model (4.1) this wavefunction can bechosen to depend on the coordinates of N -extended tensorial superspace (Xαβ, θαI), onthe bosonic spinor (s-vector) variable λα and on a half of the fermionic variables χI as far asthey are, by (4.21), their own momenta. The separation of a half of the real χI coordinatescan be achieved by introducing complex3 variables ηq, (ηq)∗ = ηq, q = 1, . . . ,N /2, so thatχI = (χq, χN/2+q) = ((ηq + ηq), i(ηq − ηq)),

ηq =χq − iχN/2+q

2, ηq =

χq + iχN/2+q

2, ηq, ηpPB = −iδqp . (4.24)

Then, the wavefunction superfield in the coordinates representation depends only on ηq,

W =W(Xαβ, θα;λα; ηq) , (4.25)

the various momenta are given by the differential operators

Pαβ = −i∂αβ , παI = −i ∂

∂θαI, ηq =

∂

∂ηq, (4.26)

3A separation in pairs of conjugate constraints is used in the Gupta-Bleuler method of quantizing systemswith second class constraints as the massive superparticle [188–190].

53

4.2. Quantization of the superparticle in Σ(n(n+1)

2|Nn) with even N and conformal higher

spin equations

and the Poisson brackets become quantum commutators or anticommutators ([ , PB 7→1i~ [ , ; we take ~ = 1). The quantum constraints operators, to be denoted by the same

symbol (although having in mind DclassicalαI 7→ −iDquantum

αI , Pclassicalαβ 7→ −iPquantumαβ ) arethen

DαI :=∂

∂θαI+ i∂αβθ

βI − χIλα , (4.27)

Pαβ := ∂αβ − iλαλβ , (4.28)

and have to be imposed on the wavefunction (4.25).

For even N , it is convenient to introduce complex Grassmann coordinates and complexGrassmann derivatives,

Θαq =1

2(θαq − iθα(q+N/2)) = (Θα

q )∗ ⇔ ∂αq :=∂

∂Θαq=

∂

∂θαq+ i

∂

∂θα(q+N/2),(4.29)

q = 1, . . . ,N /2, and conjugate pairs of fermionic constraints

∇αq := Dαq + iDα(q+N/2) = ∂αq + 2i∂αβΘβq − 2λα

∂

∂ηq= Dαq − 2λα

∂

∂ηq, (4.30)

∇αq := Dαq − iDα(q+N/2) = ∂α

q + 2i∂αβΘβq − 2λα ηq = Dαq − 2λαη

q . (4.31)

Since Dαq, Dpβ = 4i∂αβδpq , the above ∇αq, ∇α

q and the bosonic constraint (4.28)determine the superalgebra given by the only nonzero bracket

∇αq, ∇pβ = 4iPαβδpq . (4.32)

This shows that it is sufficient to impose on the superwavefunction (4.25) the fermionicconstraints,

∇αqW := DαqW − 2λα∂

∂ηqW = 0 , (4.33)

∇pαW := DqαW − 2λα η

qW = 0 , (4.34)

since the mass-shell-like bosonic constraint,

PαβW := (∂αβ − iλαλβ)W = 0 , (4.35)

will follow as a consistency condition for (4.33), (4.34).

Decomposing the superwavefunction in a finite power series in the complex Grassmannvariable ηq,

W(X,Θq, Θq, λ, ηq) = W (0)(X,Θq, Θq, λ) +

N/2∑k=1

1

k!ηqk . . . ηq1 W (k)

q1...qk(X,Θq, Θq, λ) , (4.36)

54

Chapter 4. Conformal higher spin theory

we find that Eqs. (4.33), (4.34) imply

DαqW (0) = 2λαW(1)q , ... , DαqW (k)

q1...qk= 2λαW

(k+1)qq1...qk

, . . . ,

DαqW (N/2)q1...qN/2

= 0 , (4.37)

and

DqαW (0) = 0 , DqαW (k)q1...qk

= 2kλαW(k−1)[q1...qk−1

δqk]q , . . . ,

DqαW (N/2)q1...qN/2

= NλαW (N/2−1)[q1...qN/2−1

δqqN/2] . (4.38)

Eqs. (4.37) show that all the superfields W(k)q1...qk can be constructed from fermionic derivatives

of the superfield W (0)(X,Θq, Θq, λ) which is chiral as a consequence of the first equationin (4.38),

Dαkqk . . .Dα1q1W(0) = 2kλα1 . . . λαkW

(k)q1...qk

, DqαW (0) = 0 . (4.39)

Then, the wavefunctionW is completely characterized by the chiral superfield W (0)(X,Θq, Θq, λ).As a consequence of (4.35), W (0) obeys

PαβW (0) := (∂αβ − iλαλβ)W (0) = 0 , (4.40)

which is solved by a planewave in tensorial space,

W (0)(X,Θq, Θq, λ) = W (λ,Θq, Θq) expiλαλβXαβ . (4.41)

The chirality of W (0) and the first equation in (4.37), which now implies ∂αqW(0) ∝ λα,

show that the general solution for the superparticle wavefunction is determined by thefollowing chiral plane wave superfield

W (0)(X,Θq, Θq, λ) = w(λ , Θqλ) expiλαλβ(X + 2iΘpΘp)αβ , (4.42)

where Θqλ = Θα qλα and

w(λ , Θqλ) = w(0)(λ) +

N/2∑k=1

1

k!(λΘqk) . . . (λΘq1)w(k)

q1...qk(λ) . (4.43)

We refer to [97] for a discussion on how the arbitrary function w(0)(λα) with α = 1, 2, 3, 4encodes all the solutions of the massless higher spin equations in D = 4 and to [97, 103] forthe D = 6, 10 cases. The key point is that λα carries the degrees of freedom of a light-likemomentum (λγaλ is light-like in D=4,6,10 which corresponds to n=4,8,16) plus thoseof spin. The d.o.f. of λα and those of the lightlike momenta are encoded, both up to ascale factor, in the coordinates of the compact manifolds Sn−1 = S2D−5 and S

n2 = SD−2,

respectively. The spheres4 Sn2−1 are related to helicity in the n = 4, D = 4 case and to its

4The ‘celestial spheres’ SD−2 are the bases S2,4,8 of the Hopf fibrations S2D−5 → SD−2 (Sn−1 → Sn2 )

55

4.3. From the superfield to the component form of the higher spin equations in tensorialspace

multidimensional generalizations for D = 6, 10 [97, 103].

To obtain a superfield on tensorial superspace describing massless conformal higher spintheories with extended supersymmetry, the wavefunction (4.42) has to be integrated overRn − 0 ∼ Sn−1× R+ , parametrized by λα, with an appropriate measure that we denoteby dnλ,

Φ(X,Θq, Θq) =

∫dnλW (0)(X,Θq, Θq, λ) =

∫dnλw(λ , Θqλ)eiλαλβ(X+2iΘpΘp)αβ .

(4.44)

One can easily check that the superfield Φ is chiral

DqαΦ(X,Θq′ , Θp′) = 0 (4.45)

and satisfies the equation

Dq[βDγ]pΦ(X,Θq′ , Θp′) = 0 . (4.46)

These are the superfield equations for the wavefunction of the superparticle in N -extendedtensorial superspace for even N .

4.3. From the superfield to the component form of thehigher spin equations in tensorial space

4.3.1. N = 2

When N = 2, Eqs. (4.45) and (4.46) can be written as

DαΦ = 0 ,

D[αDβ]Φ = 0 (4.47)

and reproduce equations from page XIV. It is easy to check that Eqs. (4.47) imply thevanishing of all the components of the ‘chiral’ superfield Φ(X,Θ, Θ) = Φ(X + 2iΘ · Θ,Θ),except the first two,

Φ(X,Θ, Θ) = φ(XL) + iΘαψα(XL) , XαβL = Xαβ + 2iΘ(αΘβ) = Xβα

L , (4.48)

where XαβL is the analogue of the bosonic coordinates for the chiral basis of standard

D = 4 superspace. The above components are the complex bosonic scalar and the complex

of S 3,7,15, (n,D)=(4,4), (8,6), (16,10). The fibres SD−3 = S1,3,7 of these bundles correspond to thecomplex, quaternion and octonion numbers of unit modulus. The remaining n=2, D=3 case correspondsto the first of the four Hopf fibrations, S1 → RP 1; its fibre is determined by the reals of unit modulus,Z2, and there are no extra coordinates.

56

Chapter 4. Conformal higher spin theory

fermionic spinor fields obeying the free higher spin equations in tensorial space form [98],

∂α[γ∂δ]βφ(X) = 0 , ∂α[βψγ](X) = 0 . (4.49)

Let us recall that the N = 1 supermultiplet contains a real bosonic scalar and a real fermionicspinor field that obey the same equations (4.49). Hence, the N = 2 supermultiplet ofthe conformal fields in tensorial superspace is given by the complexification of the N = 1supermultiplet.

Clearly, the above results are n-independent and thus, besides n = 4, they are also validfor the n = 8 and n = 16 cases corresponding to the D = 6 and D = 10 multiplets ofmassless conformal higher spin fields.

4.3.2. N = 4

In contrast with the N = 2 case, spin-tensorial components are present when N > 2. ForN = 4, the general solution of the superfield equations (4.45) and (4.46) is given by

Φ(X,Θq, Θq) = φ(XL) + iΘαqψαq(XL) + εpqΘαqΘβpFαβ(XL) , (4.50)

XαβL = Xαβ + 2iΘq(αΘβ)

q , q = 1, 2 , (4.51)

where, again, the complex scalar and spinor fields obey the standard higher spin equationsin their tensorial superspace form,

∂α[γ∂δ]βφ(X) = 0 , ∂α[βψγ]q(X) = 0 , (4.52)

while the complex symmetric bi-spinor (or ‘s-tensor’ [98]) Fαβ = Fβα satisfies the tensorialcounterpart of the D = 4 Maxwell equations (when these are written in spinorial notation[176], see also below),

∂α[γFδ]β(X) = 0 , Fαβ = Fβα . (4.53)

However, one can easily show that the general solution of Eq. (4.53) is expressed througha new complex scalar superfield φ(X) satisfying the bosonic tensorial space equation in(4.52),

Fαβ = ∂αβφ(X) , (4.54)

∂α[γ∂δ]βφ(X) = 0 . (4.55)

n = 4 , D = 4

To prove this when n = 4 (α, β = 1, 2, 3, 4), we begin by decomposing the complexsymmetric GL(4) tensor Fαβ = Fβα in 2 × 2 blocks, thus keeping only the GL(2,C)

57

4.3. From the superfield to the component form of the higher spin equations in tensorialspace

manifest symmetry5,

n = 4 : Fαβ =

(FAB VABVBA FAB

), A,B = 1, 2 , A, B = 1, 2 . (4.56)

Let us first notice that the block components of Eq. (4.53) which contain the antisymmetrictensors (encoded in the symmetric spin-tensors FAB and FAB) only,

∂A[BFC]D = 0 , ∂A[BFC]D = 0 , (4.57)

are equivalent to the Maxwell equations for the complex selfdual field Fab = i2εabcdF

cd ∝σab

CDFCD i.e., they imply ∂aFab = 0 and ∂[aFbc] = 0.

Consider now the components of Eq. (4.53) which contain the complex vector VAB =σaABVa only, namely

∂A[BVC]D = 0 , ∂B[AVD]C = 0 (4.58)

and

∂A[BVC]D = 0 , ∂A[BVC]D = 0 . (4.59)

Eqs. (4.58) imply ∂[aVb] = 0 and ∂aVa = 0. The first is solved by Va = ∂aφ and the second

implies that the scalar field φ obeys the Klein-Gordon equation ∂a∂aφ = 0. In the spin-tensornotation these read

VAB = ∂ABφ , ∂A[B∂C]Dφ = 0 . (4.60)

Next, the components of Eq. (4.53) which contain both vector and antisymmetric tensorcomponents, ∂ABFCD − ∂ACVDB = 0 and ∂ABFCD − ∂CDVAB = 0, can be written in theform

∂AB(FCD − ∂CDφ) = 0 , ∂AB(FCD − ∂CDφ) = 0 , (4.61)

the only covariant solution of which is given by

FCD = ∂CDφ , FCD = ∂CDφ . (4.62)

Keeping in mind the Maxwell equations (4.57), one finds that the scalar field φ(X) satisfies,besides the Klein-Gordon equation in (4.60), also the remaining components of Eq. (4.55),

∂A[B∂C]Dφ = 0 , ∂A[B∂C]Dφ = 0 . (4.63)

5Notice that the SL(2,C) indices A,B = 1, 2, A, B = 1, 2 are denoted by α, β and α, β in chapters 2and 3.

58

Chapter 4. Conformal higher spin theory

Proof for arbitrary n

We now prove that Eqs. (4.54), (4.55) provide the general solution of the Maxwell-likeequation in tensorial space, Eq. (4.53), for any n. The Fourier transform of Eq. (4.53) is

pα[γFδ]β(p) = 0 . (4.64)

The solution of this equation is nontrivial iff the matrix of the generalized momentum hasrank one, this is to say when pαβ = λαλβ for arbitrary λα 6= (0, ..., 0) or, equivalently, whenthis matrix obeys pα[γpδ]β = 0. The general solution is characterized by Fαβ(λ) = λαλβφ(λ)

and can be equivalently written in the form Fαβ(p) = pαβφ(p) if pα[γpδ]βφ(p) = 0 6 ofEqs. (4.53). As far as set of equations

Fαβ(p) = pαβφ(p) , pα[γpδ]βφ(p) = 0 (4.65)

provide the Fourier transforms of Eqs. (4.54), (4.55), these describe the general solution.

Peccei-Quinn-like symmetry

Thus, the N = 4 higher spin supermultiplet actually contains two complex scalar fields andtwo Dirac spinor fields in tensorial space, φ(X), ψ1

α(X), ψ2α(X), φ(X), which satisfy the

free bosonic and fermionic higher spin equations, Eqs. (4.52), (4.55). They appear in theon-shell scalar superfield decomposition as

Φ(X,Θq, Θq) = φ(XL) + iΘαqψαq(XL) + εpqΘαqΘβp∂αβφ(XL) , q, p = 1, 2 . (4.66)

However, as the second complex scalar field φ enters the original superfield with a derivative,its zero mode is not fixed. In other words, this scalar is axion-like: it possesses thePeccei-Quinn-like symmetry

φ(X) 7→ φ(X) + const . (4.67)

4.3.3. N = 8

For higher N > 4 the general solution of the set of superfield equations (4.45) and (4.46)is given by

Φ(X,Θq, Θq) = φ(XL) + iΘαqψαq(XL) +

N/2∑k=2

1

k!Θαkqk . . .Θα1q1Fα1...αk q1...qk(XL) , (4.68)

Fα1...αk q1...qk(XL) = F(α1...αk) [q1...qk](XL) , XαβL = Xαβ + 2iΘq(αΘβ)

q , q = 1, ...,N /2 , (4.69)

6More formally, the solution of this equation is given by a distribution with support on the subspace oftensorial momentum space defined by the condition pα[γpδ]β = 0, so that φ(p) ∝ δ(pα[γpδ]β).

59

4.3. From the superfield to the component form of the higher spin equations in tensorialspace

where φ(XL) and ψαq(XL) obey the standard higher spin equations (4.49) while the highercomponents satisfy

∂α[γFδ]β2...βk q1...qk(XL) = 0 , Fα1...αq = F(α1...αq) . (4.70)

For instance, for N = 8 the superfield solution of the higher spin equations (4.45) and(4.46) reads

Φ(X,Θq, Θq) = φ(XL) + iΘαqψαq(XL) +1

2Θα2q2Θα1q1Fα1α2 q1q2(XL) +

+i

3!Θα3q3Θα2q2Θα1q1εq1q2q3qψ

qα1α2α3

(XL) +

+1

4!εq1q2q3q4Θ

α4q4 . . .Θα1q1Fα1...α4(XL) . (4.71)

Its two lowest components obey Eqs. (4.49), while its higher order nonvanishing fieldcomponents satisfy Eqs. (4.70),

∂α[γFδ]β q1q2(X) = 0 , (4.72)

∂α[γψδ]β2β3q(X) = 0 , (4.73)

∂α[γFδ]β2β3β4(X) = 0 . (4.74)

It is tempting to identify (4.73) with the tensorial space generalization of the Rarita-Scwingerequations and Eq. (4.74) with that of the linearized conformal gravity equation imposed onWeyl tensor. However, similarly to the N = 4 case in Sec. 4.3.2, it is possible to show thatthe general solutions of Eqs. (4.72), (4.73) and (4.74) are expressed in terms of a sextupletof scalar fields φq1q2(X) = φ[q1q2](X), a quadruplet of spinorial fields ψqα3

and a singlet of

scalar field φ(X) obeying the standard tensorial space fermionic and bosonic higher spinequations (4.49),

Fαβ q1q2(X) = ∂αβφq1q2(X) , (4.75)

ψqα1α2α3(X) = ∂α1α2ψ

qα3

(X) , (4.76)

Fα1...α4(X) = ∂α1α2∂α3α4φ(X) . (4.77)

Summarizing, the N = 8 supermultiplet of free higher spin fields is described by a setof two scalar fields, a sextuplet of scalar fields, a spinor field and a quadruplet of spinorialfields, all in tensorial superspace, which obey the usual type free higher spin equations

∂α[γ∂δ]βφ(X) = 0 , ∂α[βψγ](X) = 0 ,

∂α[γ∂δ]βφqp(X) = 0 ,

∂α[βψγ]q(X) = 0 , ∂α[γ∂δ]βφ(X) = 0 . (4.78)

Thus, we conclude that, in tensorial superspace, at least all the free field dynamics is carriedby the scalar and spinor fields. However, the ‘higher’ scalar and spinor fields appear in thebasic superfield under the action of one or two derivatives and, hence, the model is invariant

60

Chapter 4. Conformal higher spin theory

under the following generalized bosonic and fermionic Peccei-Quinn-like symmetries

φqp(X) 7→ φqp(X) + aqp ,

ψαq(X) 7→ ψα

q(X) + βαq ,

φ(X) 7→ φ(X) + a+Xαβaαβ , (4.79)

with constant bosonic parameters aqp = −apq, a, aαβ and constant fermionic parameterβα

q. Note that the non-constant shift in φ(X) is allowed by the presence of two derivativesin Eq. (4.77).

61

CHAPTER 5

COVARIANT ACTION AND EQUATIONSOF MOTION FOR THE 11D SYSTEM OF

MULTIPLE M0-BRANES

In this chapter we present and study the covariant supersymmetric and κ–symmetricaction for a system of N nearly coincident M0–branes (mM0 system) in flat elevendimensional superspace and the equations of motion obtained from this action. As

far as the mM0 action is written with the use of moving frame and spinor moving framevariables, we begin by describing their use in a simpler model of single M0–brane. We obtainthe complete set of mM0 equations of motion and study the symmetries of the mM0 actionthe most relevant of which is the reminiscence of the K9 symmetry of the single M0–brane.This symmetry allows us to write the bosonic equations of motion for the mM0 centerof energy variables in their final form. The center of energy motion is characterized by anonnegative constant mass M which is contructed from the matrix fields which describethe relative motion of the mM0 constituents. We show that all supersymmetric solutions ofthe mM0 equations preserve 16 of 32 supersymmetries i.e. describe 1

2BPS states, and are

characterized by vanishing center of energy mass M2 = 0. We also present two examples ofnon supersymmetric solutions with M2 6= 0.

63

5.1. Single M0–brane in spinor moving frame formulation

5.1. Single M0–brane in spinor moving frameformulation

5.1.1. Twistor–like spinor moving frame action and its irreducibleκ–symmetry

The spinor moving frame action of M0–brane reads (see [191] and also [192–195])

SM0 =

∫W 1

ρ# E= =

∫W 1

ρ# u=a E

a(Z) (5.1)

= 1/16

∫W 1

ρ# (v −q Γav−q ) Ea . (5.2)

In the first line of this equation, (5.1), ρ#(τ) is a Lagrange multiplier,

Ea := Ea(Z) = dZM(τ)EaM(Z) =: dτEa

τ (Z) (5.3)

is the pull–back of the bosonic supervielbein of the 11D target superspace (a = 0, 1, ..., 10),Ea = Ea(Z) = dZMEa

M(Z), to the worldline W 1 parametrized by proper time τ . In thecase of flat target superspace the supervielbein can be chosen in the form 1,2

Ea = Πa = dxa − idθΓaθ , Eα = dθα. (5.4)

Finally, E= = Eau=a and u=

a = u=a (τ) is a light–like 11D vector, u=au=

a = 0.

One can write the action (5.1) in a probably more conventional from, extracting dτmeasure from the pull–back of the supervielbein 1–form (see (5.3))

SM0 =

∫W 1

dτρ# E=τ =

=

∫W 1

dτρ# ∂τ ZM(τ)Ea

M(Z(τ))u=a (τ) . (5.5)

We however, prefer to hide dτ inside of differential form, define the Lagrangian 1-form byL1 = dτLτ , and write our actions as integral of this 1–form over the worldline,

∫W 1 L1,

rather than as an integral over dτ of a density,∫dτLτ .

If we were stoping at this stage, one can easily observe that the action (5.1) can be

1The action (5.1), (5.2) makes sense when supervielbein Ea = dZMEaM (Z) obeys the 11D superspacesupergravity constraints [149]. In this chapter we will restrict ourselves by the case of flat targetsuperspace, described by Eqs. (5.4).

2We use the (real) matrices Γaαβ = Γaβα = ΓaαγCγβ and Γαβa = Γβαa = CαγΓaγ

β constructed as a product

of 11D Dirac matrices Γaβγ (obeying ΓaΓb + ΓbΓa = 2ηabI32×32) with, respectively, the 11D charge

conjugation matrix Cγβ = −Cβγ and its inverse Cαβ = −Cβα. Both Γaβγ and Cβγ are pure imaginary

in our mostly minus notation ηab = diag(1,−1, ...,−1).

64

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

obtained from the first order form of 11D version of the Brink–Schwarz action,

SBS =

∫W 1

(paE

a − e

2pap

adτ), (5.6)

by solving the constraints papa = 0 (equations of motion for Lagrange multiplier e(τ)) and

substituting them back to the action. Furthermore, one might wonder why the solutionpa = ρ#u=

a is written with a multiplier ρ#(τ) instead of just stating that it has the form ofS =

∫W 1 paE

a with pa constrained by papa = 0. We will answer that question a bit later,

just announcing now that ρ# is a kind of Stuckelberg variable allowing to introduce anSO(1, 1) gauge symmetry; although looking artificial at this stage, this symmetry allows toclarify the group theoretical meaning of u=

a and also of the set of 16 constrained spinorsappearing in the second representation of SM0, Eq. (5.2).

The light–like vector u=a can be considered as a composite of (any of) the 16 spinors v−αq

provided these are constrained by

v−αq (Γa)αβv−βp = δqpu

=a (5.7a)

2v−αq v−βq = u=a Γaαβ . (5.7b) (5.7)

Notice that the trace of (5.7a) as well as the Γ–trace of (5.7b) give 16u=a = v−αq (Γa)αβv

−βq

which can be read off (5.2) and (5.1). The set of spinors v−αq constrained by (5.7) are calledspinor moving frame variables (hence the name ‘spinor moving frame’ for the formulation ofsuperparticle mechanics based on the action (5.1), (5.2)). Before discussing their origin andnature (in sec. 5.1.3), in the next sec. 5.1.2 we would like to try to convince the reader inthe usefulness of these ’square roots’ of the light–like vector u=

a .

5.1.2. Irreducible κ–symmetry of the spinor moving frame action

The action (5.1), (5.2) is invariant under the following local fermionic κ–symmetry transfor-mations

δκxa = −iθΓaδκθ , δκθ

α = ε+q(τ)v−αq ,

δκρ# = 0 ,

δκu=a = 0 ⇐ δκv

−αq = 0 . (5.8)

These symmetry is irreducible in the sense of that each of 16 fermionic parameters3 ε+q(τ)acts efficiently on the variables of the theory and can be used to remove some component offermionic field θα(τ) thus reducing the number of the degrees of freedom in it to 16 (whileα = 1, ..., 32).

3The κ–symmetry was discovered in [188, 196] and was shown to coincide with the local worldlinesupersymmetry in [197]. Our notation ε+q(τ) for the (irreducible) κ–symmetry parameter is an implicitreference on this later result which will be useful in the discussion below.

65

5.1. Single M0–brane in spinor moving frame formulation

In contrast, the κ–symmetry of the original Brink–Schwarz superparticle action (5.6) [196]

δκxa = −iθΓaδκθ , δκθ

α = paΓaαβκβ(τ) ,

δκe = −4iκβdθβ , (5.9)

is infinitely reducible. It is parametrized by 32 component fermionic spinor function κβ(τ)which however is not acting efficiently on the variable of the theory.4

The irreducible κ–symmetry of the spinor moving frame formulation (5.8) can be obtainedfrom the infinitely reducible (5.9) by substituting for pa the solution pa = ρ#u=

a of theconstraint pap

a = 0; furthermore, using (5.7), we find

ε+q = 2ρ#v−αq κα . (5.10)

Let us stress that this relation, as well as the transformation rules of the irreducible κ–symmetry (5.8), necessarily involves the constrained spinors v−αq . Thus the covariantirreducible form of the κ–symmetry is a characteristic property of the spinor moving frameand similar (’twistor–like’) formulations of the superparticle mechanics.5

The importance of the κ–symmetry is related to the fact that it reflects a part of targetspace supersymmetry which is preserved by ground state of the brane under consideration[71, 77] thus insuring that it is a BPS state. Its irreducible form, reached in the frame ofspinor moving frame formulation, is useful not only for clarifying its nature as worldlinesupersymmetry ([197]), but also for finding the corresponding induced supergravity multipletwhich is necessary for constructing the mM0 action. To address this issue we need tocomment on some properties of moving frame and spinor moving frame variables.

5.1.3. Moving frame and spinor moving frame

To clarify the origin and nature of the set of spinors v−αq which provide the square root ofthe light–like vector u=

a in the sense of Eqs. (5.7), and which have been used to present theκ–symmetry in the irreducible form (5.8), it is useful to complete the null–vector u=

a till themoving frame matrix,

U(a)b =

(u=b +u#b

2, uib,

u#b −u=b

2

)∈ SO(1, 10) (5.11)

4Roughly speaking, due to the constraint papa = 0, κα and κα + paΓaαβκ

(1)β(τ) produce the same κvariation of the Brink–Schwarz superparticle variables. One says that the above transformation has anull-vector κ(1)β(τ) and, hence, the symmetry is reducible. But this is not the end of story. One easily

observes that κ(1)β(τ) and κ(1)β(τ) + paΓaαβκ(2)β (τ), with an arbitrary κ

(2)β (τ), makes the same change

of the parameter κα. This implies that there is a null–vector for null–vector and that the κ–symmetrypossesses at least the second rank of reducibility. Furthermore, one sees that this process of finding highernull–vectors can be continued up to infinity (next stages are completely equivalent to the first two ones)so that one speaks about infinite reducibility of the κ–symmetry of the Brink–Schwarz superparticle. Thenumber of the fermionic degrees of freedom which can be removed by κ–symmetry is then calculated asan infinite sum 32−32+32−32+... = 32·(1−1+1−1+...) = 32· lim

q→1(1−q+q2−...) = 32· lim

q→1

11+q = 16.

5Notice that in D=3,4 and 6 dimensions the counterpart of v−αq can be chosen to be unconstrainedspinors; see references in e.g. [191, 192, 194].

66

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

(i = 1, ..., 9). The statement that this matrix is an element of the SO(1, 10), having beenmade in (5.11), is tantamount to saying that

UTηU = I , ηab = diag(+1,−1, ...,−1) , (5.12)

which in its turn implies that the moving frame vectors obey the following set of constraints[198]

u=a u

a = = 0 , u=a u

a i = 0 , u =a u

a# = 2 , (5.13)

u#a u

a# = 0 , u #a u

ai = 0 , (5.14)

uiauaj = −δij. (5.15)

The 11D spinor moving frame variables (appropriate for our case) can be defined as 16× 32blocks of the Spin(1, 10) valued matrix

V α(β) =

(v+αq

v−αq

)∈ Spin(1, 10) (5.16)

double covering the moving frame matrix (5.11). This statement implies that the similaritytransformations with the matrix V leaves the 11D charge conjugation matrix invariant and,when applied to the 11D Dirac matrices, produce the same effect as 11D Lorentz rotationwith matrix U ,

V CV T = C , (5.17)

V ΓbVT = U

(a)b Γ(a) , (5.18)

V T Γ(a)V = ΓbU(a)b . (5.19)

The two seemingly mysterious constraints (5.7) appear as a 16×16 block of the second ofthese relations, (5.17), and as a component V T Γ=V = Γbu=

b of the third one, (5.19) (withan appropriate representation of the 11D Gamma matrices (see Appendix E)). The otherblocks/components of these constraints involve the second set of constrained spinors,

v+q Γav

+p = u#

a δqp , v−q Γav+p = −uiaγiqp , (5.20)

2v+αq v+

qβ = Γaαβu#

a , 2v−(αq v+

qβ) = −Γaαβuia . (5.21)

Here γiqp are the 9d Dirac matrices; they are real, symmetric, γiqp = γipq, and obey theClifford algebra

γiγj + γjγi = 2δijI16×16 , (5.22)

67

5.1. Single M0–brane in spinor moving frame formulation

as well as the following identities

γiq(p1γip2p3) = δq(p1δp2p3) , (5.23)

γijq(q′γip′)p + γijp(q′γ

ip′)q = γjq′p′δqp − δq′p′γ

jqp . (5.24)

Thus v−αq and v+αq can be identified as square roots of the light–like vectors u=

a and u#a ,

respectively, while to construct uia one needs both these sets of constrained spinors.

The first constraint, Eq. (5.17), implies that the inverse spinor moving frame matrix

V (β)α =

(vαq

+ , vαq−) ∈ Spin(1, 10) , (5.25)

V(β)γV (α)

γ = δ(β)(α) =

(δqp 00 δqp

)⇔

v−αq vαp

+ = δqp = v+αq vαp

− ,

v−αq vαp− = 0 = v+α

q vαp+ ,

can be constructed from v∓αq ,

vα−q = iCαβv

−βq , vα

+q = −iCαβv+β

q . (5.26)

5.1.4. Cartan forms, differentiation and variation of the (spinor)moving frame variables

To vary the action and to clarify the structure of the equations of motion one needs tovary and to differentiate the moving frame and spinor moving frame variables. As these areconstrained, at the first glance this problem might look complicated, but, actually, this isnot the case. The clear group theoretical structure beyond the moving frame and spinormoving frame variables makes their differential calculus and variational problem extremelysimple.

Referring again for the details to [139, 191], let us just state that the derivatives ofthe moving frame and spinor moving frame variables can be expressed in terms of theso(1, 10)–valued Cartan forms Ω(a)(b) = U (a)cdU

(b)c the set of which can be split onto the

covariant Cartan forms

Ω=i = u=aduia , Ω#i = u#aduia , (5.27)

providing the basis for the coset SO(1,10)SO(1,1)×SO(9)

, and the forms

Ω(0) =1

4u=adu#

a , (5.28)

Ωij = uiaduja , (5.29)

which have the properties of the SO(1, 1) and SO(9) connection respectively. These canbe used to define the SO(1, 1)× SO(9) covariant derivative D. The covariant derivative of

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Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

the moving frame vectors is expressed in terms of the covariant Cartan forms (5.27)

Dub= := dub

= + 2Ω(0)ub= = ub

iΩ=i , (5.30)

Dub# := dub

# − 2Ω(0)ub# = ub

iΩ#i , (5.31)

Dubi := dub

i − Ωijubj =

1

2ub

#Ω=i +1

2ub

=Ω#i . (5.32)

The same is true for the spinor moving frame variables,

Dv−αq := dv−αq + Ω(0)v−αq −1

4Ωijγijqpv

−αp =

= −1

2Ω=iv+α

p γipq , (5.33)

Dv+αq := dv+α

q − Ω(0)v+αq −

1

4Ωijγijqpv

+αp =

= −1

2Ω#iv−αp γipq . (5.34)

The variation of moving frame and spinor moving frame variables can be obtained from theabove expression for derivatives by a formal contraction with variation symbol, iδd = δ (thisis to say, by taking the Lie derivatives). The independent variations are then described by iδcontraction of the Cartan forms, iδΩ

(a)(b). Furthermore, iδΩ(0) and iδΩ

ij are the parametersof the SO(1, 1) and SO(9) transformations, which are manifest gauge symmetries of themodel. Then the essential variation of the moving frame and spinor moving frame variables,this is to say, variations which produce (better to say, which may produce) nontrivialequations of motion, are expressed in terms of iδΩ

=i and iδΩ#i,

δub= = ub

iiδΩ=i , δub

# = ubiiδΩ

#i , (5.35)

δubi =

1

2ub

#iδΩ=i +

1

2ub

=iδΩ#i . (5.36)

δv−αq = −1

2iδΩ

=iv+αp γipq , (5.37)

δv+αq = −1

2iδΩ

#iv−αp γipq . (5.38)

5.1.5. K9 gauge symmetry of the spinor moving frame action ofthe M0–brane

A simple application of the above formulae begins by observing that the parameter iδΩ#i

does not enter the variation of neither u=a nor v−αq . However, the M0–brane (5.1), (5.2)

involves only these (spinor) moving frame variables. Hence the transformation of the spinormoving frame corresponding to τ dependent parameters k#i = iδΩ

#i are gauge symmetries

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5.1. Single M0–brane in spinor moving frame formulation

of this M0 action. These so–called K9–symmetry transformations

δub= = 0, δub

# = ubik#i, δub

i =1

2ub

=k#i , (5.39)

δv−αq = 0, δv+αq = −1

2k#iv−αp γipq (5.40)

should be taken into account when calculating the number of M0 degrees of freedom.

Quite interesting remnants of this K9 symmetry survives in the multiple M0 case and willbe essential to understand the structure of mM0 equations of motion.

5.1.6. Derivatives and variations of the Cartan forms

One can easily check that the covariant Cartan forms are covariantly constant,

DΩ=i = 0 , DΩ#i = 0 , (5.41)

where the covariant derivatives include the induced connection (5.28), (5.29) 6. Thecurvatures of these connections,

F (0) := dΩ(0) =1

4Ω= i ∧ Ω# i , (5.42)

Gij := dΩij + Ωik ∧ Ωkj = −Ω= [i ∧ Ω# j] , (5.43)

can be calculated, e.g., from the integrability conditions of Eqs. (5.30)–(5.32),

DDu#a = 2F (0)u#

a , DDuai = ujaG

ji . (5.44)

As in the case of moving frame variables (see sec. 5.1.4), the variations of the Cartan formscan be obtained from the above expressions using the Lie derivative formula. Omitting thetransformations of manifest gauge symmetries SO(1,1) and SO(9) (parametrized by iδΩ

(0)

and iδΩij), we present the essential variations:

δΩ#i = DiδΩ#i , δΩ=i = DiδΩ

=i , (5.45)

δΩij = Ω=[iiδΩ#j] − Ω#[iiδΩ

=j] , (5.46)

δΩ(0) =1

4Ω=iiδΩ

#i − 1

4Ω#iiδΩ

=i . (5.47)

These equations will be useful to vary the multiple M0–brane action in Sec. 5.4. For derivingthe equations of motion of single M0–brane it is sufficient to use Eqs. (5.35), (5.37) and(5.30)–(5.34).

6DΩ=i := dΩ=i + 2Ω=i ∧ Ω(0) − Ω=j ∧ Ωji, see (5.30)–(5.32).

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Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

5.2. Equations of motion of a single M0–brane andinduced N = 16 supergravity on the worldline W 1

The moving frame matrix U(b)a (5.11) provides a ‘bridge’ between the 11D Lorentz group

and its SO(9)⊗ SO(1, 1) subgroup in the sense that it carries one index (a) of SO(1, 10)and one index ((b)) transformed by a matrix from SO(9)⊗ SO(1, 1) subgroup of SO(1, 10).Contracting the pull–back of the bosonic supervielbein form Eb we arrive at

E(a) = EbU(a)b = (E=, E#, Ei) (5.48)

which is split covariantly in three types of one forms. These are inert under SO(1, 10) butcarry the nontrivial SO(9) vector index (in the case of Ei) or SO(1, 1) weights (in thecases of E= and E#). The corresponding decomposition of the vector representation ofSO(1, 10) with respect to its SO(9)⊗ SO(1, 1) subgroup,

11 7→ 1−2 + 1+2 + 90 ,

is even better illustrated by the equation E(a)U(a)b = Eb which, in more detail, reads

Ea =1

2E=ua# +

1

2E#ua= − Eiuai . (5.49)

Thus the moving frame vectors help to split the pull–back of the supervielbein in a Lorentzcovariant manner. The SO(9) singlet one form with SO(1, 1) weight -2, E= = Ebu=

b entersthe action (5.1) multiplied by the weight +2 worldline scalar field ρ#(τ). This clearly hasthe meaning of the Lagrange multiplier: its variation results in vanishing of E=,

E= := Eau=a = 0 . (5.50)

Now, the variation of E= contain two different contributions, δE= = δEau=a + Eaδu=

a . Thefirst comes from the variation of the pull–back of the bosonic supervielbein form which inour case of flat target superspace can be easily calculated with the result

δEa = −idθΓaδθ + d(δxa − iδθΓaθ) . (5.51)

The second term contains the variation of the light–like vector u=a which can be written

as in Eq. (5.35), δu=a = uia iδΩ

=i with an arbitrary iδΩ=i. The corresponding variation

of the action (5.1) reads δuSM0 =∫W 1 ρ

# δu=a E

a =∫W 1 ρ

# uia EaiδΩ

=i and produce theequation of motion

Ei := Eauia = 0 . (5.52)

Using Eq. (5.49) one can collect Eqs. (5.50) and (5.52) in

Ea :=1

2E#ua= . (5.53)

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5.2. Equations of motion of a single M0–brane and induced N = 16 supergravity on theworldline W 1

This equation shows that the M0–brane worldline W 1 is a light–like line in target (super)space,as it should be for the massless superparticle.

Furthermore (5.53) suggests to consider E# as einbein on the worldline W 1; this compositeeinbein is induced by embedding of W 1 into the target superspace. The transformation ofE# under the irreducible κ–symmetry (5.8) is given by δκE

# = −2iE+qε+q. In the light ofthe identification of κ–symmetry with local worldline supersymmetry [197], this equationsuggests to consider the covariant 16+ projection, E+q = Eαv+q

α , of the pull–back of thefermionic 1–form Eα as induced ‘gravitino’ companion of the induced 1d ‘graviton’ E#.Indeed under the κ–symmetry (5.8) this set of forms show the typical transformations rulesof (1d N = 16) supergravity multiplet,

δκE+q = Dε+q(τ) , δκE

# = −2iE+qε+q . (5.54)

Here D = dτDτ is the SO(1, 1)× SO(9) covariant derivative which we will specify below.The connection in this covariant derivative are defined in terms of moving frame variables and,hence, are inert under the κ–symmetry; in this sense the induced 1d N = 16 supergravitymultiplet is described essentially by 1 bosonic and 16 fermionic 1–forms E# and E+q. Ouraction for the mM0 system, which we present in the next section, will contain the couplingof these induced 1d supergravity to the matter describing the relative motion of the mM0constituents.

The other, 16− projection E−q = Eαv−qα of the pull–back of fermionic supervielbein formto W 1 vanishes on the mass shell,

E−q := Eαv−qα = 0 . (5.55)

Indeed, varying the coordinate functions in the action (5.1) we arrive at equation δSM0

δZM= 0

which reads

∂τ (ρ#u=

aEaM(Z)) = 0 . (5.56)

In our case of flat target superspace EaM(Z) = δaM − iδαM(Γaθ)α and one can easily split

(5.56) into the bosonic vector and fermionic spinor equations (which we prefer to write withthe use of d = dτ∂τ )

d(ρ#u=a ) = 0 , (5.57)

ρ#u=a (Γa∂τ θ)α = 0 . (5.58)

Using (5.7b) and assuming ρ# 6= 0 we find that (5.58) is equivalent to Eq. (5.55). Thisimplies that the dθα can be expressed through the induced gravitino,

Eα = dθα = E+qv−αq . (5.59)

Let us come back to the equation for the bosonic coordinate functions, (5.57) (orequivalently, ∂τ (ρ

#u=a ) = 0). Using (5.30) we can write this in the form 0 = Dρ# u=

a +ρ#uiaΩ

=i. Here and below we use the covariant derivatives defined in (5.30), (5.31), (5.32)).

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Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

Contracting that equation with ua# gives us

Dρ# = 0 , (5.60)

while the nontrivial part of the bosonic equations of motion of a single M0–brane, which canbe read off from the coefficient for uia, states that the covariant Cartan form Ω=i vanishes,

Ω=i = 0 . (5.61)

Coming back to Eq. (5.30), we see that Eq. (5.61) can be expressed by stating that thecovariant derivative of the light–like vector u=

a vanishes,

Du=a = 0 , (5.62)

or, equivalently, by

Dv−αq = 0 . (5.63)

On the other hand, using

D = dτDτ = E#D# , (5.64)

we can write Eq. (5.62) in the form D#u=a = 0, and, as far as (5.53) implies u=

a = 2Ea#, in

the following more standard form

D#Ea# = 0 , (5.65)

or, in more detail,

D#D#xa = iD#(D#θΓ

aθ) . (5.66)

Two more observations will be useful below. The first is that Eq. (5.60), 0 = Dρ# =dρ# − 2ρ#Ω(0), can be solved with respect to the induced SO(1, 1) connection,

Ω(0) =dρ#

2ρ#. (5.67)

Notice that this is in agreement with the statement that one can always gauge away any 1dconnection: using the local SO(1,1) symmetry to fix the gauge ρ# = const we arrive atΩ(0) = 0.

The second comment concerns the supersymmetric pure bosonic solutions of the aboveequations of motion.

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5.3. Covariant action for multiple M0–brane system

5.2.1. All supersymmetric solutions of the M0 equations describe1/2 BPS states

As far as the fermionic coordinate function θα is transformed by both spacetime supersym-metry and by the worldline supersymmetry (κ–symmetry), δθα = −εα + ε+q(τ)v−αq (τ), thepurely bosonic solutions of the M0 equations, having

θα = 0 , (5.68)

may preserve a part of target space supersymmetry. This is characterized by parameter

εα = ε+q(τ)v−αq (τ) . (5.69)

The left hands side of this equation contains a constant fermionic spinor dεα = 0, sothat d(ε+qv−αq ) = Dε+qv−αq + ε+qDv−αq = 0. Furthermore, taking into account that theequations of motion for the bosonic coordinate function, Eq. (5.66), implies (5.63), onefinds that the consistency of (5.69) is the covariant constancy of the κ–symmetry parameterε+q(τ),

Dε+q = 0 . (5.70)

In 1d system the connection can be gauged away so that this condition can be reducedto the existence of a constant SO(9) spinor εq. For instance gauging away the SO(9)

connection and using Eq. (5.67), we can present (5.70) in the form d(ε+q/√ρ#) = 0 and

solve it by ε+q =√ρ# εq with dεq = 0.

This implies that any purely bosonic solution of the M0 equations preserves exactly 1/2of the spacetime supersymmetry.

5.3. Covariant action for multiple M0–brane system

5.3.1. Variables describing the mM0 system

Let us introduce the dynamical variables describing the system of multiple M0–branes, whichwe abbreviate as mM0. Its dimensional reduction is expected to produce the system of Nnearly coincident D0-branes (mD0 system) and at very low energy this later is described bythe action of 1d N = 16 supersymmetric Yang–Mills theory (SYM) with the gauge groupU(N), which is given by dimensional reduction of the 10D N = 1 U(N) SYM down tod=1. Now, the set of fields of the U(N) SYM can be split onto the non-Abelian SU(N)SYM and Abelian U(1) SYM multiplets. Roughly speaking, this later describes the centerof energy motion of the mD0 system while the former corresponds to the relative motion ofthe constituents of the mD0 system. Then it is natural to assume that the relative motionof the mM0 constituents are also described by the fields of SU(N) SYM multiplet.

Now let us turn to the center of energy motion. We begin by noticing that the U(1)SYM fields can be seen in the single D0 brane action (see [65] and refs therein) after fixingthe gauge with respect to κ–symmetry and reparametrization symmetry. Originally the

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Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

action of a single D0 brane is written in terms of 10 bosonic and 32 fermionic coordinatefunctions, worldline fields corresponding to the coordinates of type IIA D = 10 superspace.The above gauge fixing reduces the number of fermionic fields to 16 and the number ofbosonic coordinate functions to 9. These are the same as the number of physical fields as in1d reduction of the 10D SYM theory. This also contains the time component of the gaugefield which can be gauged away by the U(1) gauge symmetry transformation and do notcarry degrees of freedom. The U(1) SYM multiplet describing the center of energy motionof the mD0 system can be obtained by fixing the gauge with respect to κ-symmetry andreparametrization symmetry on the coordinate functions, the same as in the case of singleD0 brane.

In the light of the above discussion, it is natural to describe the center of energy motionof the mM0 system by the 11 bosonic and 32 fermionic coordinate functions, the sameas used to describe the motion of single M0–brane, and to assume that the wanted mM0action possesses κ–symmetry and reparametrization symmetry, like the single M0–braneaction does.

To resume, following [139, 146, 150] we will describe the center of energy motion of Nnearly coincident M0–branes (mM0 system) by the 11 commuting and 32 anti-commutingcoordinate functions

ZM(τ) = (xµ(τ), θα(τ)) , (5.71)

µ = 0, 1, ..., 10; α = 1, 2, ..., 32

(the same as used to describe single M0–brane), and the relative motion of the mM0constituents by the fields of the SU(N) SYM supermultiplet. These are the bosonic andfermionic Hermitian traceless N ×N matrices fields

Xi(τ) and Ψq(τ) (5.72)

(i = 1, ..., 9 , q = 1, ..., 16)

depending on a (center of energy) proper time variable τ . The bosonic Xi(τ) carries theindex i = 1, ..., 9 of the vector representation of SO(9), while the fermionic Ψq transformsas a spinor under SO(9), q = 1, ..., 16.

5.3.2. First order form of the 1d N = 16 SYM Lagrangian as astarting point to build mM0 action

The standard 1d N = 16 SYM Lagrangian (obtained by dimensional reduction of 10D SYM)can be written in the following first order form

dτLSYM = tr(−Pi∇τXi + 4iΨq∇τΨq

)+ dτH (5.73)

where the Hamiltonian

H =1

2tr(PiPi

)+ V(X)− 2 tr

(Xi ΨγiΨ

)(5.74)

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5.3. Covariant action for multiple M0–brane system

contains the positively definite scalar potential

V = − 1

64tr[Xi,Xj

]2 ≡ +1

64tr[Xi,Xj

]·[Xi,Xj

]†, (5.75)

Eqs. (5.73) and (5.74) involve the auxiliary ‘momentum’ fields, the nanoplet of tracelessN ×N matrices Pi, and also the gauge field Aτ (τ) which enters the covariant derivatives∇ = dτ∇τ of the above bosonic and fermionic fields,

∇Xi = dXi + [A,Xi] , ∇Ψq = dΨq + [A,Ψq] . (5.76)

The action with the above Lagrangian are invariant under the following d=1 N = 16supersymmetry transformations with constant fermionic parameter εq

δεXi = 4iεq(γiΨ)q , δεPi = [εq(γiΨ)q,Xj] , (5.77)

δεΨq =1

2εpγipqPi −

i

16εpγijpq[Xi,Xj] , (5.78)

δεA = −dτεqΨq . (5.79)

The mM0 action should describe the coupling of the above SYM theory to the centerof energy variables (5.71). As we have discussed above, such an action should possess thereparametrization symmetry and a 16 parametric local fermionic symmetry, a counterpartof the irreducible κ–symmetry (5.8) of the single M0 action. It is natural also to thinkon this fermionic gauge symmetry as on the local version of the above rigid d=1 N = 16supersymmetry of the SYM action, Eqs. (5.77)–(5.79).

5.3.3. Induced supergravity on the center of energy worldline

The natural way to make a supersymmetry local is to couple it to supergravity multiplet.As a by–product such a coupling should guaranty the reparametrization (general coordinate)invariance. Now it is the time to recall about induced supergravity multiplet on the worldlineof the single M0–brane constructed in sec. 5.2. Similarly, we can associate a movingframe (5.11) and spinor moving frame (5.16) to the center of energy motion of the mM0system and use these together with center of energy coordinate functions (5.71) to build thecomposite d=1 N = 16 supergravity multiplet including the 1d ‘graviton’ and ‘gravitino’

E# = Eau#a = (dxa − idθΓaθ)u#

a , (5.80)

E+q = Eαv+qα = dθαv+q

α , (5.81)

transforming under the local supersymmetry as in (5.54),

δεE+q = Dε+q(τ) , δεE

# = −2iE+qε+q . (5.82)

Notice that the use of such a composite supergravity induced by embedding of the centerof energy worldline into the flat target 11D superspace implies that the local supersymmetryparameter carries the weight +1 of the SO(1, 1) group transformations defined on the

76

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

moving frame variables. This implies the necessity to adjust the SO(1, 1) weight also to thefields describing the relative motion of the mM0 constituents. Following [139, 146, 150] wedefine the SO(1, 1) weight of the bosonic and fermionic fields to be -2 and -3, respectively,so that in a more explicit notation (and using the conventions were the upper − indexindicate the same -1 weight as the lower + one) 7

Xi = Xi# := Xi

++ , i = 1, ..., 9 , (5.83)

Ψq = Ψ# +q := Ψ++ +q = Ψ#−q , q = 1, ..., 16 . (5.84)

As in the case of single M0–brane, we expect the SO(1, 1) as well as SO(9) transforationto be a gauge symmetry of our action. This implies the use of covariant derivative withSO(1, 1) and SO(9) connection. As in the case of single M0–brane, we define theseconnections to be constructed from the moving frame variables

Ω(0) =1

4u=adu#

a , Ωij = uiaduja (5.85)

(see Eqs. (5.28) and (5.29)), which are now associated to the center of energy motion ofthe mM0 system. The covariant derivatives of the su(N) valued matrix fields (5.83) aredefined by

DXi := dXi + 2Ω(0)Xi − ΩijXj + [A,Xi] , (5.86)

DΨq := dΨq + 3Ω(0)Ψq −1

4ΩijγijqpΨp + [A,Ψq] . (5.87)

They also involve the SU(N) connection A = dτAτ (τ) on the center of energy worldlineW 1. The anti-Hermitian traceless N × N matrix gauge field Aτ (τ) is an independentvariable of our model. Let us stress, however, that, as any 1d gauge field, it can be gaugedaway and thus does not carry any degree of freedom.

The covariant derivative of the supersymmetry parameter in (5.82) reads

Dε+q = dε+q − Ω(0)ε+q +1

4Ωijε+pγijpq , (5.88)

so that the induced connection (5.85) are also the members of the composite d=1 N = 16supergravity multiplet.

5.3.4. A way towards mM0 action

Now we are ready to present the action for the system of N nearly coincident M0–branes(mM0 system) which was proposed in [150]. It can be considered as a result of ‘gauging’of rigid d=1 N = 16 supersymmetry (5.77)–(5.79) of the SU(N) SYM action with the

7Such a chose of weight of the basic matrix fields is preferable for the description in the frame ofsuperembedding approach, like developed in [139, 146]. Once using the density ρ# = ρ++ which entersthe spinor moving frame action for single M0, we can easily change the weights of the fields multiplyingthem by corresponding power of ρ#. However we find more convenient to work with the ‘weighted’fields (5.83), (5.84).

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5.3. Covariant action for multiple M0–brane system

Lagrangian (5.73) achieved by coupling it to a composite d=1 N = 16 supergravity (5.80),(5.81), (5.85) induced by embedding of the center of energy worldline of the mM0 systeminto the target 11D superspace.

The natural first step on this way is to make the Lagrangian (5.73) covariant by couplingit to a 1d gravity. This can be reached by just replacing dτ in the right hand side of (5.73) bythe 1-form E# of (5.80). Then, to provide also the SO(1, 1) and SO(9) gauge symmetries,which play the role of Lorentz and R-symmetries in our induced 1d N = 16 supergravity,we should replace the Yang–Mills covariant derivatives in (5.76) by the SO(1, 1)× SO(9)covariant derivatives defined in (5.86), (5.87), and to multiply the Lagrangian 1-form thusobtained by (ρ#)3. The next stage is suggested by the fact that setting N=1 in the actionfor the system of N nearly coincident M0–brane one should arrive a single M0–brane action.As the SU(N) SYM Lagrangian, and all the matrix fields involved in it, vanish when N=1,this implies the necessity just to add the single M0 action to the integral of the abovedescribed Lagrangian form. Then the coupling to induced gravitino can be restored fromthe requirement of local supersymmetry invariance of the mM0 action.

5.3.5. mM0 action

In such a way we arrive at the mM0 action proposed in [150]. It reads

SmM0 =

∫W 1

ρ# E= +

+

∫W 1

(ρ#)3(tr(−PiDXi + 4iΨqDΨq

)+ E#H

)+

+

∫W 1

(ρ#)3 E+qtr

(4i(γiΨ)qPi +

1

2(γijΨ)q[Xi,Xj]

),

(5.89)

where H is the relative motion Hamiltonian (cf. (5.74))

H := H####(X,P,Ψ) =

=1

2tr(PiPi

)+ V(X)− 2 tr

(Xi ΨγiΨ

)(5.90)

including the scalar potential (cf. (5.75))

V := V####(X) = − 1

64tr[Xi,Xj

]2(5.91)

= +1

64tr[Xi,Xj

]·[Xi,Xj

]†, (5.92)

and the Yukawa–type coupling tr (Xi ΨγiΨ).

The covariant derivatives D are defined in (5.86), (5.87). Their connection are build fromthe (spinor) moving frame variables, Eq. (5.85), which are related to the center of energymotion of the mM0 system. These are also used to construct the composite graviton and

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Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

gravitino 1-forms E# and E+q, Eqs. (5.80), (5.81). The 1-form E= is the same as in thecase of single M0–brane

E= = Eau=a . (5.93)

For the completeness of this section, let us recall that in these equations Ea is the pull–back of the bosonic supervielbein to the center of energy worldline W 1, Eq. (5.3), (5.4),Eα = dθα(τ), u=

a and u#a are light–like moving frame vectors (5.11), (5.13), (5.14), and

v+qα is an element of spinor moving frame (5.16).

Although the first term in (5.89) coincides with the single M0–brane action (5.1), nowthe Lagrange multiplier ρ# and spinor moving frame variables are also present in the secondand third terms. This results in that their equations of motion differ from (5.53), and, aswe discuss in the next section, generically, the center of energy motion of the mM0 systemis not light-like.

5.3.6. Local supersymmetry of the mM0 action

The action (5.89) is invariant under the transformation of the 16 parametric local worldlinesupersymmetry

δεθα = ε+q(τ)v−αq , (5.94)

δεxa = −iθΓaδεθ +

1

2ua#iεE

= , (5.95)

δερ# = 0 , (5.96)

δεv±αq = 0 ⇒ δεu

=a = δεu

#a = δεu

ia = 0 , (5.97)

δεXi = 4iε+γiΨ , δεPi = [(ε+γiΨ),Xj] , (5.98)

δεΨq =1

2(ε+γi)qPi −

i

16(ε+γij)q[Xi,Xj] , (5.99)

δεA = −E#ε+qΨq + E+γiε+ Xi , (5.100)

where

iεE= = 6(ρ#)2tr

(iPiε+γiΨ− 1

8ε+γijΨ[Xi,Xj]

). (5.101)

The local supersymmetry transformations of the fields describing relative motion of mM0constituents, (5.98), (5.99) coincide with the SYM supersymmetry (5.77), (5.78) modulothe fact that now the fermionic parameter is an arbitrary function of the center of energyproper time, ε+q = ε+q(τ). The local supersymmetry transformation of the 1d SU(N)gauge field (5.100) differs from the SYM transformation by additional term involving thecomposite gravitino.

The transformations of the center of energy variables Eqs. (5.94)–(5.97) describe adeformation of the irreducible κ–symmetry (5.8) of the free massless superparticle. Actually,the deformation touches the transformation rule (5.95) for the the bosonic coordinatefunction, δεx

a only. The Lagrange multiplier ρ# and the (spinor) moving frame variables

79

5.4. mM0 equations of motion

are invariant under the supersymmetry, like they are under the κ-symmetry of a singlesuperparticle.

5.4. mM0 equations of motion

In this section we present and study the complete set of equations of motion for the multipleM0–brane system which follow from the action (5.89).

5.4.1. Equations of the relative motion

Varying the action with respect to the momentum matrix field Pi gives us the equation

DXi = E#Pi + 4iE+q(γiΨ)q (5.102)

which allows to identify Pi, modulo fermionic contribution, with the covariant time derivativeof Xi,

Pi = D#Xi − 4iE+#γ

iΨ . (5.103)

Here

D# =1

E#τ

Dτ , E+q# =

1

E#τ

E+qτ , (5.104)

are covariant derivative and the induced 1d gravitino field corresponding to the inducedeinbein on the worldvolume, E# = Eau#

a =: dτE#τ , in the sense of that

D = E#D# , E+q = E#E+q# . (5.105)

The variation with respect to the worldline gauge field A = dτAτ gives

[Pi,Xi] = 4iΨq , Ψq (5.106)

and the variation with respect to Xi results in

DPi = − 1

16E#[[Xi,Xj]Xj] + 2E# ΨγiΨ + E+qγijqp[Ψp,Xj] . (5.107)

Using (5.102) we can easily present this equation in the form

D#D#Xi = − 1

16[[Xi,Xj]Xj] + 2ΨγiΨ + 4iD#(E+

#γiΨp) + E+

#γij[Ψ,Xj] . (5.108)

Finally, the variation with respect to the traceless matrix fermionic field Ψq produces

DΨ =i

4E#[Xi, (γiΨ)] +

1

2E+γi Pi − i

16E+γij [Xi,Xj] . (5.109)

80

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

5.4.2. A convenient gauge fixing

To simplify the above equations, let us use the fact that 1-dimensional connection canalways be gauged away and fix the gauge where the composed SO(9) connection (5.29)and also the SU(N) gauge field vanish

Ωij = dτΩijτ = 0 , (5.110)

A = dτAτ = 0 . (5.111)

This breaks the local SO(9) and SU(N), but the symmetry under the rigid SO(9)⊗SU(N)transformations remains.

As far as the SO(1, 1) gauge symmetry is concerned, we would not like to fix it but ratheruse a part 1

2ua# δSmM0

δxa= 0 of the equations of motion for the center of energy coordinate

functions xa (discussed below in full),

Dρ# = 0 , (5.112)

to find the explicit form of the induced SO(1, 1) connection (5.28), Ω(0) := 14ua=du#

a .Indeed, as far as Dρ# = dρ# − 2ρ#Ω(0), Eq. (5.112) implies

Ω(0) =dρ#

2ρ#. (5.113)

In the gauge (5.110), (5.111) the set of bosonic gauge symmetries is reduced to theAbelian SO(1, 1), τ–reparametrization and b–symmetry (which we describe below in sec.5.4.5), and the covariant derivatives simplify to

DXi = (ρ#)−1d(ρ#Xi) ,

DPi = (ρ#)−2d((ρ#)2Pi) ,DΨq = (ρ#)−3/2d((ρ#)3/2Ψq) . (5.114)

As a result, Eqs. (5.108) and (5.109) can be written in the following (probably moretransparent) form:

∂τ Ψ =i

4e [Xi, (γiΨ)] +

1

2√ρ#E+τ γ

i Pi − i

16√ρ#E+τ γ

ij [Xi, Xj] , (5.115)

∂τ

(1

e∂τ Xi

)= − e

16[[Xi, Xj], Xj] + 2 e ΨγiΨ +

+4i∂τ

(E+τ γ

iΨ

e√ρ#

)+

1√ρ#E+τ γ

ij[Ψ, Xj] . (5.116)

81

5.4. mM0 equations of motion

Writing Eqs. (5.115) and (5.116) we used the redefined fields

Xi = ρ#Xi , Ψq = (ρ#)3/2Ψq , (5.117)

Pi = (ρ#)2Pi =1

e

(∂τ Xi − 4i√

ρ#E+τ γ

iΨ

), (5.118)

which are inert under the SO(1, 1), and

e(τ) = E#τ /ρ

# (5.119)

which has the properties of the einbein of the Brink–Schwarz superparticle action (5.6).

5.4.3. By pass technical comment on derivation of the equationsfor the center of energy coordinate functions

This is the place to present some comments on the convenient way to derive equations ofmotion for the center of energy variables (which was actually used as well when workingwith single M0 in Sec 5.2).

To find the manifestly covariant and supersymmetric invariant linear combinations of theequations of motion for the bosonic and fermionic coordinate functions, δSmM0

δxa= 0 and

δSmM0

δθα= 0, we introduce the covariant basis iδE

A in the space of variation such that

δZMSmM0 =

∫W 1

(δxa

δSmM0

δxa+ δθα

δSmM0

δθα

)=

=

∫W 1

(iδE

a δSmM0

iδEa+ iδE

α δSmM0

iδEα

). (5.120)

In the generic case of curved superspace iδEA = δZMEA

M(Z); in our case of flat targetsuperspace this implies

iδEa = δxa − iδθΓaθ , iδE

α = δθα . (5.121)

Furthermore, it is convenient to use the moving frame variables to split covariantly the setof bosonic equations δSmM0

iδEa= 0 into

δSmM0

iδE==

1

2ua# δSmM0

iδEa,

δSmM0

iδE#=

1

2ua= δSmM0

iδEa,

δSmM0

iδEi= −uai δSmM0

iδEa, (5.122)

82

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

and the set of fermionic equations, δSmM0

iδEα= δSmM0

δθα, into

δSmM0

iδE−q= v+α

q

δSmM0

δθα,

δSmM0

iδE+q= v−αq

δSmM0

δθα. (5.123)

To resume,

δZMSmM0 =

∫W 1

(δxa − iδθΓaθ)(u=a

δSmM0

iδE=+

+u#a

δSmM0

iδE#+ uia

δSmM0

iδEi

)+

+

∫W 1

δθα(v−qα

δSmM0

iδE−q+ v+q

α

δSmM0

iδE+q

).

(5.124)

5.4.4. Equations for the center of energy coordinate functions

As we have already stated, the bosonic equation δSmM0

iδE=:= 1

2ua# δSmM0

δxa= 0 results in Eq.

(5.112) which is equivalent to (5.113). This observation is useful to extract consequencesof the next equation, δSmM0

iδE#= 0, which reads

D((ρ#)3H) = 0 . (5.125)

Using (5.112) one can write Eq. (5.125) in the form of

d((ρ#)4H) = 0 . (5.126)

or, equivalently, (ρ#)4H = const. Due to the structure of H, Eq. (5.90), this constantis nonnegative. Furthermore, as it has been shown in [150] (see also sec. 5.7.3), it canbe identified (up to numerical multiplier) with the mass parameter M2 characterizing thecenter of energy motion,

M2 = 4(ρ#)4H = const ≥ 0 . (5.127)

The remaining projection of the equation for the bosonic center of energy coordinatefunctions, δSmM0

iδEi:= −1

2uai δSmM0

δxa= 0, gives us the relation between covariant SO(1,10)

SO(1,1)×SO(9)

Cartan forms (5.27),

Ω=i = −(ρ#)2H Ω#i = − M2

4(ρ#)2Ω#i . (5.128)

83

5.4. mM0 equations of motion

The nontrivial part of the fermionic equation of the center of energy motion, δSmM0

iδE−q:=

v−αqδSmM0

iδEα= 0, reads

E−q = −1

2Ω#i γiqpν

−#p , (5.129)

where

ν −#q := (ρ#)2tr

((γjΨ)qPj −

i

8(γjkΨ)q[Xj,Xk]

). (5.130)

5.4.5. Noether identities for gauge symmetries. First look.

Actually one can show that Eq. (5.125) is satisfied identically when other equations aretaken into account. (To be precise, Eqs. (5.102), (5.106), (5.107), (5.109), (5.112) have tobe used). This is the Noether identity for the ’tangent space’ copy of the reparametrizationsymmetry (sometimes it is called ’b-symmetry’) with the parameter function iδE

#. Similarly,one can find the Noether identity reflecting the existence of the N = 16 1d gaugesupersymmetry (5.94)–(5.101) with the basic parameter ε+q = iδE

+q . It states thedependence of the one half of the fermionic equations, namely δSmM0

iδE+q:= v−αq

δSmM0

iδEα= 0,

which reads

Dν −#q = ρ2E+qH (5.131)

or Dν −#q = ρ#2E+qH#### in a more complete notation.

5.4.6. Equations which follow from the auxiliary field variationsand simplification of the above equations

Variation with respect to the Lagrange multiplier ρ#, δSmM0

δρ#= 0, expresses the projection

E= := Eau=a of the pull–back Ea of the bosonic supervielbein to the center of energy

worldline through the relative motion variables,

E= := Eau=a = −3(ρ#)2L### =

= 3(ρ#)2tr

(1

2PiDXi +

1

64E#[Xi,Xj]2 − 1

4(E+γijΨ)[Xi,Xj]

). (5.132)

The Ei := Eauia projection of this pull–back is expressed by equations appearing as aresult of variation with respect to the spinor moving frame variables. According to Eqs.(5.35)–(5.38), that should appear as coefficients for iδΩ

=i and iδΩ#i in the variation of the

action. Equation δSmM0

iδΩ=i = 0 reads

Ei := Eauia = −(ρ#)−1 Ω#j(J ij + δijJ

), (5.133)

84

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

where we have introduced the notation

J ij := (ρ#)3 tr(P[iXj] − iΨγijΨ

), (5.134)

J :=(ρ#)3

2tr(PiXi

). (5.135)

The (ρ#)3 multipliers are introduced to make J ij and J inert under the SO(1, 1) transfor-mations.

In this notation, equation δSmM0

iδΩ#i = 0 reads

(ρ#)3HEi = −Ω=j(J ij − δijJ

)− 2i(ρ#)E−q(γiν−#)q .

(5.136)

Using (5.128), (5.133), (5.127) and (5.129), one can rewrite Eq. (5.136) as equation forΩ#i,

Ω#j(M2J ij − 2i(ρ#)2ν−#γ

ijν−#)

= 0 . (5.137)

Actually, as we are going to show in the next sec.5.4.7, taking into account the remnant ofthe K9 gauge symmetry of single M0–brane (see (5.39) and (5.40)) , which is present inthe mM0 action, one can present the above equation in the form of

Ω#i = 0 , (5.138)

or, in terms of component, Ω#jτ = 0. Due to (5.128) Eq. (5.138) implies

Ω=i = 0 (5.139)

and (5.133) acquires the same form as in the case of single M0–brane,

Ei := Eauia = 0 . (5.140)

Furthermore, the fermionic equation of motion (5.129) also becomes homogeneous, of thesame form as the equation for single M0–brane,

E−q = 0 . (5.141)

Eqs. (5.138) and (5.139) also imply that all the moving frame and spinor moving framevariables are covariantly constant,

Du#a = 0 , Du=

a = 0 , Duia = 0 , (5.142)

Dv+αq = 0 , Dv−αq = 0 . (5.143)

Notice that in the case of single M0–brane such a form of equations for moving framevariables can be reached after gauge fixing the K9 gauge symmetry with parameter iδΩ

#i. Inthe mM0 case only a part (remnant) of K9 symmetry is present so that a part of variations

85

5.4. mM0 equations of motion

iδΩ#i produce nontrivial equations which, together with the above mentioned remnant of

K9 symmetry, results in Eqs. (5.142), (5.143).

5.4.7. Noether identity, remnant of the K9 gauge symmetry andthe final form of the Ω#i equation

In this section we present the remnant of K9 gauge symmetry leaving invariant the mM0action and show that, modulo this gauge symmetry, Eq. (5.137) is equivalent to (5.138).

Let us write Eq. (5.137) as

Ω#jτ ijג = 0 , (5.144)

where

ijג = M2J ij − 2i(ρ#)2ν−#γijν−# . (5.145)

As this 9×9 matrix is antisymmetric, it has rank 8 or lower, rank(גij) ≤ 8. In other words,it has at least one ‘null vector’, this is to say a vector V i which obey8

∃ V i, i = 1, ..., 9 : ijVג j = 0 . (5.146)

Actually, the matrix ijג is constructed from the dynamical variables of our model in such away (according to Eqs. (5.145) and (5.134)) that the number of its null vectors dependson the configuration of the fields describing the relative motion of the mM0 constituents.However, as one ‘null vector’ always exists, it is sufficient to consider a configuration withrank(גij) = 8, and ijג having just one ‘null vector’, at some neighborhood ∆τ of a proper-time moment τ ; the generalization for a more complicated configurations/neighborhoods isstraightforward.

Then, on one hand, the solution of Eq. (5.144) in the neighborhood ∆τ is given byΩ#iτ ∝ V i, or, equivalently,

Ω#iτ = f V i , (5.147)

where f = f(τ) is an arbitrary function of the center of energy proper time τ . [Forconfigurations/neighborhoods with several ‘null vectors’ V i

r , r = 1, ..., (9 − rank J) thesolution will be Ω#i

τ = f r V ir with arbitrary functions f r = f r(τ)].

On the other hand, the existence of null vector, Eq. (5.146), implies that a part of Eqs.(5.144) is satisfied identically

Ω#jτ ijVג j ≡ 0 , (5.148)

when some other equations are taken into account. This is the Noether identity reflecting

8This should not be confused with light–like vectors which can exist in the space with indefinite metric. Inparticular, our 11D moving frame vectors u=a and u#a are light–like. To exclude any confusion, in thischapter we never use the name ’null-vectors’ for the light–like vectors.

86

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

the existence of the gauge symmetry with the basic variation9

iδΩ#i = αV i (5.149)

with an arbitrary function α = α(τ). This is clearly a remnant of the K9 gauge symmetry(5.39) of the action (5.1) for single M0–brane.

The generic variation of the Cartan 1–form Ω#i can be expressed as in Eq. (5.45), whichin our 1d case can also be written as

δΩ#iτ = Dτ iδΩ

#i . (5.150)

Applying (5.150) to the variation of the solution (5.147) of Eq. (5.144) under (5.149), wefind that

δf(τ) = ∂τα(τ) . (5.151)

Hence, one can use the local symmetry (5.149) to set f = 0 and, thus, to gauge away (totrivialize) the solution (5.147) of Eq. (5.144).

This proves that the gauge fixing version of Eq. (5.144) is given by Eq. (5.138), Ω#i = 0.

In sec. 5.7 we give more detailed discussion of the above local symmetry and its Noetheridentities reproducing independently the above conclusion for the purely bosonic case.

5.5. Ground state solution of the relative motionequations

The natural first step in studying the above obtained mM0 equations is to address the sectorof

Ψq = 0 . (5.152)

As far as the fermionic equations of motion have the same form (5.141) as for the singleM0–brane, E−q = 0, the only possible fermionic contribution to the relative motion equations

might come from the induced gravitino E+q = dθαv+qα . However, with (5.152), the fermionic

equation of the relative motion (5.109) results in

E+γi Pi − i

8E+γij [Xi,Xj] = 0 . (5.153)

As it will be clear after our discussion below, for M2 > 0 this equation has only trivialsolution E+q = 0, while for M2 = 0 the 1d gravitino E+q remains arbitrary.

9See sec. 5.7 for more details on these Noether identity and gauge symmetry in the purely bosonic case.Here let us just recall that Eq. (5.137) appears as an essential part of the coefficient for iδΩ

#i in thevariation of the mM0 action.

87

5.5. Ground state solution of the relative motion equations

5.5.1. Ground state of the relative motion

It is easy to see that a particular configuration of the bosonic fields for which Eq. (5.153) issatisfied is

Pi = 0 , [Xi,Xj] = 0 . (5.154)

Then the fermionic 1-form E+q remains arbitrary (and pure gauge) as it is in the case ofsingle M0–brane.

Together with (5.152), Eqs. (5.154) describe the ground state of the relative motion. Forit the relative motion Hamiltonian (5.90) and the center of energy effective mass vanish,

M2 = 0 (5.155)

so that the center of energy motion is light–like. Moreover, when Eqs. (5.152) and (5.154)hold, all the equations of the center of energy motion coincide with the equations for singleM0–brane.

The ground state of the mM0 system is thus described by Eqs. (5.152), (5.154) and bya (pure bosonic) ground state solution of the single M0 equations. This preserves all 16worldline supersymmetries, which corresponds (as we have discussed in sec. 5.2.1) to thepreservation of 16 of 32 spacetime supersymmetries.

5.5.2. Solutions with M 2 = 0 have relative motion in the groundstate sector

Curiously enough, being in the ground state of the relative motion is the only possibilityfor the mM0 system to have the light–like center of energy motion characterized by zeroeffective mass

M2 = 0 ⇔ H =1

2tr(PiPi)− 1

64tr[Xi,Xj]2 = 0 . (5.156)

Indeed, the pure bosonic relative motion HamiltonianH is given by the sum of two terms bothof which are traces of squares of Hermitian operators ([[Xi,Xj]† = [Xj,Xi] = −[Xi,Xj]);hence, the sum vanishes, H = 0, iff both equations in (5.154) hold 10, Pi = D#Xi = 0 and[Xi,Xj] = 0 11.

Thus any nontrivial configuration of the relative motion, with either Pi 6= 0 or/and[Xi,Xj] 6= 0, creates a nonzero effective mass of the center of energy motion, M2 6= 0.

10We do not discuss here the possible nilpotent contributions, like the possibility to solve the equationa2 = 0 for a real bosonic a(τ) by a = βα1...α17

θα1 . . . θα17 with 17 center of energy fermions θα(τ)contracted with some fermionic βα1...α17 = β[α1...α17].

11This is true for finite size matrices. In the N 7→ ∞ limit (mM0 condensate) one can consider a ’non–commutative plane’ solution with [Xi,Xj ] = iΘij and c-number valued Θij = −Θji, see for instance,[199]. In the case of finite N this solution cannot be used as far as the right hand side is assumed to beproportional to the unity matrix, IN×N while the trace of the commutator vanishes.

88

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

5.6. Supersymmetric solutions of mM0 equations

5.6.1. Supersymmetric solutions of the mM0 equations haveM 2 = 0

From Eq. (5.99) one concludes that a solution of the mM0 equations with vanishing relativemotion fermionic fields, Eq. (5.152), can be supersymmetric if

(ε+γi)qPi −i

8(ε+γij)q[Xi,Xj] = 0 . (5.157)

All the 16 worldline supersymmetries (1/2 of the target space supersymmetries) can bepreserved iff this equation is satisfied for arbitrary ε+p. This implies

γiqpPi −i

8γijqp[Xi,Xj] = 0 (5.158)

the only solution of which is given by the ground state of the relative motion, Eq. (5.154).

Thus all the bosonic solutions of mM0 equations preserving 16 supersymmetries havethe trivial relative motion sector described by Eq. (5.154) which is characterized by thelight–like center of energy motion, M2 = 0.

This suggests that M2 = 0, is the BPS condition, i.e. the necessary condition for the1/2 supersymmetry preservation. As we are going to show, this is indeed the case, and,moreover

M2 = 0 (5.159)

is the BPS equation for preservation of any part of the target space supersymmetry.

Indeed, on one hand, tracing Eq. (5.157) with γjPj and using the properties of tr we find

ε+qtr(PiPi) =i

8(ε+γjkγi)qtr(Pi[Xj,Xk]) .

On the other hand, tracing (5.157) with i8γjk[Xj,Xk] and using the Jacobi identities

[X[i[Xj,Xk]]] ≡ 0 we find

i

8(ε+γiγjk)qtr(Pi[Xj,Xk]) =

1

32(ε+qtr([Xj,Xk]2) .

Taking the sum of these two equations and using (5.106) (with fermionic fields set to zero)we find ε+qH = 0 which, using (5.127), can be written as ε+qM2 = 0,

ε+qM2 = 0 ⇐ ε+qH = 0 . (5.160)

For M2 6= 0 this implies ε+q = 0, so that the supersymmetry is broken. Thus all thesupersymmetric solutions of mM0 equation are characterized by M2 = 0.

This fact is very important: it means that the existence of our action does not imply

89

5.7. On solutions of mM0 equations with M2 > 0

the existence of a new type of supersymmetric solutions of the 11D SUGRA equations12.A BPS solution is in correspondence with the ground state of the brane or of the multiplebrane system; the ground state of mM0 system is characterized by the vanishing effectivemass and with the center of energy motion characteristic for the single M0–brane. Thusa supersymmetric solution of 11D SUGRA equations corresponding to single M-wave alsodescribe the mM0 (multiple M-wave) ground state.

5.6.2. All BPS states of mM0 system are 1/2 BPS

As we have shown, a solution of mM0 equations can preserve some part of the 16 worldlinesupersymmetries (and some part (≤ 1/2) of the target space supersymmetry) if and onlyif M2 = 0. Now, in the light of the observation in sec. 5.5.2, M2 = 0 implies that therelative motion of the mM0 constituents is in its ground state, Eq. (5.154). This has twoconsequences. Firstly, as the ground state trivially solves the Killing spinor equation (5.157),it preserves all the supersymmetries allowed by the center of energy motion. Secondly, whenthe relative motion sector is in its ground state, the center of energy sector of supersymmetricsolution is described by the same equations as the motion of single M0–brane (massless11D superparticle). Now, as we have shown in sec. 5.2.1, the supersymmetric solutions ofthese M0 equations preserve just 1/2 of the target space supersymmetry.

This proves that all the supersymmetric solutions of the equations of motion of the mM0system preserve just one half of 32 target space supersymmetries. In other words, all themM0 BPS states are 1/2 BPS.

5.7. On solutions of mM0 equations with M 2 > 0

When M2 6= 0, Eq. (5.153) has only trivial solutions. (The proof of this fact follows thestages of sec. 5.6.1). This means that (5.152) results in

E+q = 0 , (5.161)

so that, when M2 > 0, a configuration with vanishing relative motion fermion is purelybosonic.

12Although this statement can be done about the solutions preserving 1/2 of the 11D supersymmetry, as itwill be clear in a moment, it is universal as far as a supersymmetric solution of mM0 equations canpreserve only 1/2 of the tangent space supersymmetry.

90

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

5.7.1. Purely bosonic equations in the case of M 2 > 0

The complete list of nontrivial pure bosonic equations for mM0 system with nonvanishingcenter of energy mass, M2 > 0, reads

Dρ# = 0 ⇔ Ω(0) =dρ#

2ρ#, (5.162)

D#D#Xi = − 1

16[[Xi,Xj]Xj] , (5.163)

[D#Xi,Xi] = 0 , (5.164)

E= := dxau=a =

= 3E#

((ρ#)2tr(D#Xi)2 − M2

4(ρ#)2

), (5.165)

Ei := dxauia = 0 , (5.166)

Ω#i = 0 , (5.167)

Ω=i = 0 , (5.168)

where E# = dxau#a and the center of energy mass M is defined by Eq. (5.127), M2 =

4(ρ#)4H, with the relative motion Hamiltonian

H = tr

(1

2(D#Xi)2 − 1

64

[Xi,Xj

]2). (5.169)

Notice that (as we have discussed in the general case) the currents

J ij = (ρ#)3 trD#X[iXj] ,

J =(ρ#)3

2trD#XiXi (5.170)

disappear from the final form of equations when one takes into account the presence of theremnants of the K9 symmetry. As far as this statement is very important in the analysis ofthe mM0 equations, we are going to give more detail on this symmetry and gauge fixingnow.

But before let us make an observation that the current J ij is covariantly constant on themass shell (i.e. when the above equations of motion are taken into account),

DJ ij = 0 . (5.171)

In contrast, in the generic purely bosonic configuration the scalar current is not a constant,DJ = dJ 6= 0.

91

5.7. On solutions of mM0 equations with M2 > 0

5.7.2. Remnant of K9 symmetry in the bosonic limit of the mM0action and Ω#i equations

The variation of the bosonic limit of the mM0 action (5.89) can be written in the form

δSbosonic

mM0 =

∫W 1

E=iu iδΩ

#i +

∫W 1

E#iu iδΩ

=i −

−∫W 1

E ixiδEi + . . . . (5.172)

where

E=iu = M2Ei/4ρ# + Ω=j(J ij − δijJ) ,

E#iu = ρ#Ei + Ω#j(J ij + δijJ) ,

E ix = ρ#Ω=i +M2Ω#i/4ρ# , (5.173)

with J ij and J defined in (5.170) and (5.135), and dots denote the terms involving the otherbasic variations (δρ#, iδE

= etc.). Furthermore, one can rearrange the terms in (5.172) inthe following way:

δSbosonic

mM0 =

∫W 1

E#iu

(iδΩ

=i − M2

4(ρ#)2iδΩ

#i

)−

−∫W 1

E ix(iδE

i +(J ij + δijJ

)iδΩ

#j)

+

+M2

2(ρ#)2

∫W 1

dτ Ω#iτ J

ijiδΩ#j + . . . , (5.174)

In this form it is transparent that the equations of motion corresponding to the iδΩ#j

variation can be written in the form

Ω#iτ J

ij = 0 , (5.175)

which is the bosonic limit of Eq. (5.137). As we have already discussed in the generalcase, Eq. (5.175) always has a nontrivial solution as far as the antisymmetric 9× 9 matrixJ ij = −J ji always has at least one null vector, a non-zero vector V i such that V iJ ij = 0.

Each null–vector generates a nontrivial solution of (5.175), but also a gauge symmetryof the mM0 action. Indeed, as one can easily see from (5.174), the transformations withτ -dependent parameter iδΩ

#j obeying

J ijiδΩ#j = 0 , (5.176)

92

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

completed by

iδΩ=i =

M2

4(ρ#)2iδΩ

#i,

iδEi = −(J ij + Jδij)iδΩ

#j , (5.177)

leave the action invariant, δSbosonic

mM0 = 0, and, thus define the gauge symmetries of themM0 action. The transformations of Ω#i

τ under this gauge symmetry are δΩ#iτ = Dτ iδΩ

#i

(5.150). As far as in purely bosonic limit DJ ij = 0 on the mass shell (see Eq. (5.171)),

J ijDτ iδΩ#j = 0 (5.178)

is also obeyed. Furthermore, in 1d case all the connection can be gauged away so that thetransformation rules of the nontrivial solution of Eq. (5.175) can be summarized as follows

δΩ#iτ = ∂τ iδΩ

#i ,

Ω#iτ J

ij = 0

J ijiδΩ#j = 0 ,

∂τJij = 0 .

(5.179)

This form makes transparent that any nontrivial solution of Eq. (5.175) can be gauged awayusing local symmetry (5.176), (5.177). Thus, modulo the gauge symmetry, Eq. (5.175) isequivalent to Eq. (5.167), Ω#i = 0.

5.7.3. Center of energy velocity and momentum for M 2 6= 0

Let us notice one property of the center of energy motion of our M0 system which, on thefirst glance, might looks strange, and try to convince the reader that it is rather a naturalmanifestation of the influence of relative motion on the center of energy dynamics.

Using Eqs. (5.165), (5.166) we can easily calculate center of energy velocity of thebosonic limit of our mM0 system,

˙xa := ∂τ xa =

1

2E=τ u

#a +1

2E#τ u

=a − Eiτu

ia =

=1

2E#τ

(u=a + 3u#a

((ρ#)2tr(D#Xi)2 − M2

4(ρ#)2

)).

(5.180)

On the other hand, the canonical momentum conjugate to the center of energy coordinatefunction ˙xa is13

pa =∂LmM0

τ

∂xa= ρ#

(u=a + u#

a

M2

4(ρ#)2

). (5.181)

This equation justifies our identification of the constant M2 as a square of the effective

13LmM0τ is the Lagrangian of the mM0 action (5.89), SmM0 =

∫dτLmM0

τ .

93

5.7. On solutions of mM0 equations with M2 > 0

mass of the mM0 system as it gives

papa = M2 . (5.182)

Thus, generically, the center of energy velocity and its momentum are oriented in differentdirections of 11D spacetime,

˙xa ∝ (pa −Aa) , (5.183)

Aa = u#a

(M2

ρ#− 3(ρ#)3tr(D#Xi)2

). (5.184)

Eq. (5.183) might look strange if one expects the center of energy motion to be similarto the motion of a free particle. However, this relation is characteristic for a charged particlemoving in a background Maxwell field (see e.g. [200]). In our case the counterpart (5.184)of the electromagnetic potential Aa is constructed in terms of the relative motion variables.It vanishes when the relative motion is in its ground state.

Thus the seemingly unusual effect of that the mM0 center of energy velocity andmomentum are not parallel one to another is just one of the manifestations of the mutualinfluence of the center of energy and the relative motion in mM0 system. The relativemotion variables, when they are not in ground state, generate a counterpart of the 11Dbackground vector potential for the center of energy motion.

5.7.4. An example of non-supersymmetric solutions

Let us fix the gauge (5.110), (5.111), Ωij = 0 = A, use the SO(1, 1) gauge symmetry toset ρ# = 1 and the reparametrization symmetry to fix E#

τ = 114,

Ωijτ = 0 = Aτ , E#

τ = 1 = ρ# . (5.185)

Then

D# = ∂τ (5.186)

and Eqs. (5.163) simplify to

Xi = − 1

16[[Xi,Xj]Xj] , (5.187)

[Xi,Xi] = 0 . (5.188)

These very well known equations describe the 1d reduction of the 10D SU(N) Yang-Millsgauge theory.

14Actually, to be precise, there exists an obstruction to fix such a gauge by τ reparametrization [201]. Thebest what one can do is to fix ∂τ E

#τ = 0, while the constant value remains indefinite. This is especially

important for path integral quantization, where the integration over this constant value (mudulus)should be included in the definition of the path integral measure. As here we do not need in this levelof precision, we allow ourselves to simplify the formulas by just setting this indefinite constant to unity.

94

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

A very simple solution of Eqs. (5.187) and (5.188) is provided by

Xi(τ) = (Aiτ +Bi)Y, (5.189)

where Y is a constant traceless N ×N matrix, Ai and Bi are constant SO(9) vectors, andτ is the proper time of the mM0 center of energy. The center of energy effective mass isdefined by the trace of Y2 and by the length of vector ~A = Ai,

M2 = 4H = 2 ~A2trY2 , ~A2 := AiAi . (5.190)

Actually, by choosing the initial point of the proper time, τ 7→ τ − a, we can always makethe constant SO(9) vectors Ai and Bi orthogonal,

~A~B := AiBi = 0. (5.191)

Then the ‘currents’ (5.170) read

J ij = A[iBj]trY2 =A[iBj]

2 ~A2M2, J =

τ

4M2 . (5.192)

Now the equations for the center of energy coordinate functions (5.132), (5.166) and thegauge fixing condition E#

τ = 1 imply

˙xau=a = 3M2/4, (5.193)

˙xauia = 0 , (5.194)˙xau#

a = 1 . (5.195)

With our gauge fixing, Eqs. (5.142), which follow from (5.167), (5.168), implies that movingframe vectors are constant

u#a = 0 , u=

a = 0 , uia = 0 . (5.196)

Thus (5.193), (5.194), (5.195) is a simple system of linear differential equations

˙x= = 3M2/4, (5.197)˙xi = 0 , (5.198)˙x# = 1 , (5.199)

for the variables

x= = xau=a , x# = xau#

a , xi = xauia . (5.200)

This system can be easily solved for the ’comoving frame’ coordinate functions (5.200). The

95

5.7. On solutions of mM0 equations with M2 > 0

solution in an arbitrary frame

xµ(τ) = xµ(0) +τ

2

(u=µ +

3M2

4u#µ

)(5.201)

describe a time-like motion of the center of energy characterized by a nonvanishing effectivemass (5.190). The velocity of this motion,

xµ =1

2

(u=µ +

3M2

4u#µ

)(5.202)

is not parallel to the canonical momentum (see (5.181))

pµ = u=µ +

M2

4u#µ . (5.203)

As it was discussed in general case in sec. 5.7.3, this is due to the influence of the relativemotion of the mM0 constituents on the center of energy motion and can be considered asan effect of the induction by the relative motion dynamics of a counterpart of the Maxwellbackground field interacting with the center of energy coordinate functions. In the caseunder consideration this induced Maxwell field is constant, Aµ = −u#

µ M2/2.

5.7.5. Another non-supersymmetric formal solution

In the case of the system of 2 M0–branes, the 2× 2 matrix field Xi can be decomposed onPauli matrices, Xi = f iJ(τ)σJ ,

σIσJ = δIJI2×2 + iεIJKσK , I, J,K = 1, 2, 3. (5.204)

The simplest ansatz which solves the Gauss constraint (5.188) is f iJ(τ) = δiJf(τ) so that

Xi(τ) = f(τ)δiJσJ , i = 1, ..., 9; I, J,K = 1, 2, 3. (5.205)

Eq. (5.187) then implies that this function should obey

f +1

2f 3 = 0 . (5.206)

The simplest solution of this equation is given by f(τ) = 2iτ

which is complex and thusbreaks the condition that Xi is a Hermitian matrix. Actually one can consider this solution,

Xi(τ) =2i

τδiJσ

J , J = 1, 2, 3. (5.207)

as an analog of instanton as far as after Wick rotation τ 7→ iτ restores the Hermiticityproperties.

Ignoring for a moment the problem with Hermiticity we can calculate the Hamiltonianand find that it is equal to zero. Thus (5.207) is a solution with vanishing center of energy

96

Chapter 5. Covariant action and equations of motion for the 11D system of multipleM0-branes

mass, M2 = 0.A configuration (5.205) with nonzero effective center of energy mass can be obtained by

observing that (5.206) has a more general solution given by the so–called Jackobi ellipticfunction [202]. These functions obey

f 2 = −f 4/4 + C (5.208)

with an arbitrary constant C. The above discussed particular solution (5.207) of (5.206)solves (5.208) with C = 0 which suggests the relation of C with M2. Indeed, a straightfor-ward calculation shows that C = M2/12 so that the instanton–like solution of the mM0equations of relative motion is given by 2x2 matrices (5.205) with the function f(τ) obeying

f 2 =M2 − 3f 4

12. (5.209)

The set of equations for the center of energy motion includes (5.198), (5.199) and

˙x= = 3M2/4− 9(f(τ))4/2 . (5.210)

This equation can be solved numerically, but its detailed study goes beyond the scope ofthis thesis.

97

CHAPTER 6

CONCLUSIONS

In conclusion we list the main results obtained in this thesis.

1. The complete set of the superfield equations of motion for the interacting system offour dimensional supermembrane and dynamical scalar multiplet have been obtained.Our study has provided the first example of superfield equations for interacting systeminvolving matter (not supergravity) superfields and supersymmetric extended object,as well as the first set of superfield equations of motion for a dynamical superfieldsystem including supermembrane. Furthermore

a) The action of supermembrane in a chiral N = 1, D = 4 superfield backgroundhas been presented for the first time.

b) It was shown that the consistency of the interaction with dynamical scalarmultiplet requires this to be special, namely to be described by chiral superfieldof special form: expressed through the real (rather than complex) pre-potentialsuperfield.

c) The equations of motion for spacetime, component fields with supermembranecontributions have been obtained from the bulk superfield equations for thesimplest case when the bulk part of the action is given by the free kinetic term(with Kahler potential K = ΦΦ). A solution of the dynamical equations forphysical fields in the leading order on supermembrane tension have been obtained.The effects of inclusion of nontrivial superpotential and relation with knownsupersymmetric domain wall solutions have been discussed.

2. The complete set of superfield equations of motion for the interacting system ofdynamical D = 4 N = 1 supergravity and supermembrane has been obtained from thesuperspace action principle. The supermembrane model is consistent in an arbitrary off–shell minimal supergravity background. However the interaction with supemembranerequires the dynamical supergravity to be the Grisaru–Siegel–Gates–Ovrut–Waldramspecial minimal supergravity. The chiral compensator superfield of this are constructedfrom real (rather than complex) prepotential.

99

a) We have developed the Wess–Zumino type approach to this special minimalsupergravity the characteristic property of which is a dynamical generation ofthe cosmological constant.

b) To see this effect in the interacting system we extract the spacetime componentequations from the superfield equations. To this end we have fixed the usual Wess–Zumino gauge in the superfield supergravity equations, supplemented by partialgauge fixing of the local supersymmetry on the supermembrane worldvolume(θα(ξ) = 0). We have shown that the supermembrane current superfields simplifydrastically in this ”WZθ=0 gauge”.

c) In the component form of equations obtained in this way it is seen that in theinteracting system the supermembrane produces a kind of renormalization of thecosmological constant, making its value different in the branches of spacetimeseparated by the supermembrane worldvolume.

d) This allowed us to show that configuration describing a domain wall separatingtwo branches of AdS space with different cosmological constants provides asupersymmetric solution of the system of our superfield supergravity equationsconsidered outside the supermembrane worldvolume W 3.

3. We have obtained the superfield equations in N = 2, 4 and 8 extended tensorialsuperspaces Σ(10|N4), which describe the supermultiplets of the D = 4 masslessconformal free higher spin field theory with N -extended supersymmetry.

a) The N = 2 supermultiplet of massless conformal higher spin equations is simplygiven by the complexification of the N = 1 supermultiplet.

b) For N = 4, 8 no tensorial space generalizations of the Maxwell, Rarita-Schwingeror linearized conformal gravity equations appear. It is shown that N ≥ 4supermultiplets are built from the scalar and spinor fields in tensorial space whichobey the standard higher spin equations in their tensorial space version. However,some of these appear in the basic superfields under derivatives, so that theN ≥ 4 supersymmetric theory is invariant under Peccei–Quinn–like symmetriesshifting these fields.

4. We have obtained and studied the equations of motion of multiple M0–brane system(mM0) which follow from the covariant supersymmetric and κ–symmetric mM0 actionproposed in [150].

a) We have found that the mM0 action is invariant under an interesting reminiscentof the so–called K9 gauge symmetry which is necessary to find the final form ofthe bosonic equations of motion for the center of energy coordinate functions.

b) We have found that, generically, there exists the ’backreaction’, the influence ofthe relative motion on the motion of the center of energy, the most importanteffects of which are that the generic center of energy motion of mM0 system ischaracterized by a nonvanishing effective mass M constructed from the matrixfield describing the relative motion. Furthermore, when the relative motion is not

100

Chapter 6. Conclusions

in its ground state, the center of energy velocity and the canonical momentumconjugate to the center of energy coordinate function are oriented in differentdirections of the 11D spacetime. These two effects disappear when M2 = 0.

c) We have shown that all the mM0 BPS states are 1/2 BPS and have the sameproperties as BPS states of single M0–brane. In particular, the effective massof the center of energy motion vanishes for the BPS states. In other words, allthe supersymmetric purely bosonic solutions of mM0 equations preserve just 1

2

of 11D supersymmetry, have the relative motion sector in its ground state sothat all the equations of the center of energy motion acquire the same form asequations for single M0–brane.

101

APPENDIX A

NOTATION, CONVENTIONS AND SOMEUSEFUL FORMULAE IN D = 4

In chapters 2 3 we use mostly minus Minkowski metric ηab = diag(1,−1,−1,−1) andcomplex Weyl spinor notation. D = 4 vector and spinor indices are denoted by symbolsfrom the beginning of Latin and Greek alphabets, a, b, c = 0, 1, 2, 3, α, β, γ = 1, 2,

α, β, γ = 1, 2. In particular, the coordinates of flat D = 4, N = 1 superspace Σ(4|4) aredenoted, respectively, by xa and θα, θα = (θα)∗. The contraction the spinorial indices

are raised and lowered by the unit antisymmetric tensors εαβ = −εβα = iσ2 ≡(

0 1−1 0

),

εαβ = −εβα and their inverse εαβ = −εβα, εαβ = −εβα. In flat superspace (or within the WZ

gauge) this implies θα = εαβθβ, θα = εαβθβ, etc. However, to get ∂αθβ = δβ

α simultaneouslywith ∂αθ

β = δαβ we have to assume that for the derivatives over the fermionic variables

∂α = −εαβ∂β so that, when we rise the spinorial index of the covariant fermionic derivative(1.22), we arrive at Dα := εαβDβ = −∂α − i(θσa)α∂a. In curved superspace we needto introduce world supervector indices M,N, ... The coordinates of curved superspace aredenoted by ZM = (xµ, θα) with α = 1, 2, 3, 4; clearly beyond the WZ guage these α are notspinorial indices so we do not use their splitting on α and ˙α (which may however, be usefulin the prepotential approach, see [151]). The star superscript ∗ denotes complex conjugationof bosonic variables and involution on Grassmann algebra (see [164] and refs. therein); in

practical terms this implies that (θαθβ)∗ = ˆθβ θα = −θα ˆθβ. Then, to keep the plus signin (θ2)∗ = θ2 with (θ)2 := θαθα, we have to define θ2 := θαθ

α. The consistency of theGrassmann algebra involution requires that (∂α)∗ ≡

(∂∂θα

)∗= −∂α ≡ − ∂

∂θα. The covariant

spinor derivative defined in (1.22) are related by (Dα)∗ = −Dα; DD := DαDα = (DD)∗

where DD := DαDα.

103

The list of properties of relativistic Pauli matrices σaβα = εβαεαβσaβα include

σaσb = ηab + i2εabcdσcσd , σaσb = ηab − i

2εabcdσcσd , (A.1)

σab := 12(σaσb − σbσa) = i

2εabcdσcd , (A.2)

σab := 12(σaσb − σbσa) = − i

2εabcdσcd , (A.3)

σabc = −iεabcdσd . (A.4)

The 3-dimensional worldvolume vector indices are denoted by symbols from the middle ofLatin alphabet. In particular, the local coordinates of the supermembrane worldvolume W 3

are denoted by ξm with m = 0, 1, 2. The worldvolume Hodge star (denoted by ∗ in the line)operation is defined as in (3.18),

∗ Ea :=1

2dξm ∧ dξn√gεmnkgklEa

l . (A.5)

In our conventions dξm ∧ dξn ∧ dξk = −εmnkd3ξ ≡ εknmd3ξ so that

∗ Ea ∧ Ea = −3d3ξ√g , ∗Ea ∧ δEa = −d3ξ

√gEmag

mnδEan (A.6)

and

δ(∗Ea ∧ Ea) = 3 ∗ Ea ∧ δEa . (A.7)

The superspace generalization of Dirac delta function reads

δ8(z) :=1

16(θ)2(θ)2δ4(x)

and obeys∫d8zδ8(z − z)f(z) = f(z) ,

∫d8zδ8(z) =

∫d4xDDDDδ8(z) = 1

104

APPENDIX B

SUPERMEMBRANE CURRENTSUPERFIELD WHICH ENTERS THE

SCALAR MULTIPLET EQUATIONS

The leading component of the Nambu-Goto current superfield (2.63)

JNG(Z) = −1

4

∫d3ξ√g

√Φ

ˆΦDDδ8(z − z)− 1

4

∫d3ξ√g

√ˆΦ

ΦDDδ8(z − z) ,(B.1)

reads

JNG|0 = +1

16

√φ

φ

∫d3ξ√g ˆθ ˆθδ4(x− x) +

1

16

√φ

φ

∫d3ξ√g θθδ4(x− x) (B.2)

The general expression for the fermionic derivative of JNG

DαJNG = −1

4

∫d3ξ√g

√Φ

ˆΦDαDDδ

8(Z − Z) . (B.3)

so that

DαJNG|0 = 1

8

√φφ

∫d3ξ√g ˆθαδ

4(x− x)− i16

∫d3ξ√g

√φˆφ

(σaθ)α(ˆθ)2 ∂aδ4(x− x) . (B.4)

Furthermore, as

DDJNG = −1

4

∫d3ξ√g

√Φ

ˆΦDDDDδ8(Z − Z) (B.5)

105

we find

DDJNG|0 =

=

∫d3ξ√g

√φ

ˆφ

(−1

4δ4(x− x)− i

4(θσa ˆθ)∂aδ

4(x− x) +1

16(θ)2(ˆθ)2δ4(x− x)

)=

= −1

4

√φ

φ

∫d3ξ√gδ4(x− x)−

− i4

√φ

φ

∫d3ξ√g(θσa ˆθ)

(∂aδ

4(x− x) +

(∂aφ

2φ− ∂aφ

2φ

)δ4(x− x)

)+O(f 4) . (B.6)

One can also write Eq. (B.6) in the equivalent but more compact form of

DDJNG|0 = −1

4

√φ

φ

∫d3ξ√g δ4(x− x) +O(f 2)[DDJNG] +O(f 4)[DDJNG] , (B.7)

where

O(f 2)[DDJNG] = − i4

√φφ

∫d3ξ√g(θσa ˆθ)

(∂aδ

4(x− x) +(∂aφ2φ− ∂aφ

2φ

)δ4(x− x)

). (B.8)

The leading term of the second, Wess–Zumino (WZ) contribution to the supermembranecurrent,

JWZ(Z) =

(2i

∫W 3

Ec ∧ Eα ∧ Eασcαα+

+1

2

∫W 3

Ec ∧ Eb ∧ EασbcαβDβ −

1

2

∫W 3

Ec ∧ Eb ∧ EασbcβαDβ−

− 1

4!

∫W 3

Ec ∧ Eb ∧ Eaεabcdσdαα[Dα, Dα]

)δ8(z − z) (B.9)

reads

JWZ(x) := JWZ(Z)|0 =1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcdθσd ˆθ δ4(x− x)−

− 1

16

∫W 3

Ec ∧ Eb ∧ dθασbcαβ θβ ˆθ ˆθ δ4(x− x)−

− 1

16

∫W 3

Ec ∧ Eb ∧ d ˆθασbcβα

ˆθβ θθ δ4(x− x) +

+i

8

∫W 3

Ec ∧ dθα ∧ d ˆθασcααθθˆθ ˆθ δ4(x− x) =

=1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcdθσd ˆθ δ4(x− x) +O(f 4) . (B.10)

106

Appendix B. Supermembrane current superfield which enters the scalar multiplet equations

Its closest fermionic partners are

DαJWZ |0 = − 1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcd(σd ˆθ)α δ

4(x− x) +

+1

16

∫W 3

Ec ∧ Eb ∧(

2(d ˆθσbcˆθ) θα + dθβσbcβα(θ)2

)δ4(x− x) +

+i

96

∫W 3

Ec ∧ Eb ∧ Eaεabcd(σdeθ)α(ˆθ)2∂eδ

4(x− x) +

+i

32

∫W 3

Ec ∧ Eb ∧ d ˆθα(σaσbc)αα(θ)2(ˆθ)2∂aδ4(x− x)−

− i4

∫W 3

Ec ∧ dθβ ∧ d ˆθβ σcββ θα (ˆθ)2 δ4(x− x) =

= − 1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcd(σd ˆθ)α δ

4(x− x) +O(f 3) . (B.11)

DαJWZ |0 =

1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θσd)α δ

4(x− x)−

− 1

16

∫W 3

Ec ∧ Eb ∧(

2(dθσbcθ)ˆθα + (d ˆθσbc)α(θ)2

)δ4(x− x)−

− i

96

∫W 3

Ec ∧ Eb ∧ Eaεabcd(ˆθσde)α(θ)2∂eδ

4(x− x)−

− i

32

∫W 3

Ec ∧ Eb ∧ (dθσbcσa)α(θ)2(ˆθ)2∂aδ

4(x− x) +

+i

4

∫W 3

Ec ∧ dθβ ∧ d ˆθβ σcββˆθα (θ)2 δ4(x− x) =

=1

48

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θσd)α δ

4(x− x) +O(f 3) . (B.12)

Then, as far as

DDJWZ(Z) = 2i

∫W 3

Ec ∧ Eα ∧ EασcααDDδ8(Z − Z) +

−1

2

∫W 3

Ec ∧ Eb ∧ ˆEασbcβα

(DβDD + 4iσa

ββ∂aD

β)δ8(Z − Z) +

+i

6

∫W 3

Ec ∧ Eb ∧ Eaεabcd∂dDDδ8(Z − Z) , (B.13)

DDJWZ(Z) = 2i

∫W 3

Ec ∧ Eα ∧ EασcααDDδ8(Z − Z) +

+1

2

∫W 3

Ec ∧ Eb ∧ Eασbcαβ(DβDD − 4iσa

ββ∂aD

β)δ8(Z − Z)−

− i6

∫W 3

Ec ∧ Eb ∧ Eaεabcd∂dDDδ8(Z − Z) , (B.14)

107

one finds that

DDJWZ |0 =1

4

∫W 3

Ec ∧ Eb ∧ d ˆθσbcˆθδ4(x− x)− i

4!

∫W 3

Ec ∧ Eb ∧ Eaεabcd(ˆθ)2∂dδ4(x− x)−

− i2

∫W 3

Ec ∧ dθα ∧ d ˆθασcαα(ˆθ)2δ4(x− x) +

+i

8

∫W 3

Ec ∧ Eb ∧ d ˆθα(θσaσbc)α (ˆθ)2∂aδ4(x− x) =

=1

4

∫W 3

Ec ∧ Eb ∧ d ˆθσbcˆθδ4(x− x)−

− i4

∫W 3

Ec ∧ Eb ∧ Eaεabcd(ˆθ)2∂dδ4(x− x) +O(f 4) . (B.15)

DDJWZ |0 =1

4

∫W 3

Ec ∧ Eb ∧ dθσbcθδ4(x− x) +i

4!

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θ)2∂dδ4(x− x)−

− i2

∫W 3

Ec ∧ dθα ∧ d ˆθασcαα(θ)2δ4(x− x)−

− i8

∫W 3

Ec ∧ Eb ∧ dθσbcσa ˆθ (θ)2∂aδ4(x− x) =

=1

4

∫W 3

Ec ∧ Eb ∧ dθσbcθδ4(x− x) +

+i

4

∫W 3

Ec ∧ Eb ∧ Eaεabcd(θ)2∂dδ4(x− x) +O(f 4) . (B.16)

To analyze the structure of the auxiliary field equation one needs also to know

[Dα , Dβ]JWZ |0 = − 1

4!

∫W 3

Ec ∧ Eb ∧ Eaεabcdσdαβδ4(x− x) +O(f 2) , (B.17)

108

Appendix B. Supermembrane current superfield which enters the scalar multiplet equations

where

O(f 2) =i

4!

∫W 3

Ec ∧ Eb ∧ Eaεabcd

((σdeθ)α

ˆθβ + θα(ˆθσde)β

)∂eδ

4(x− x) +

+1

4

∫W 3

Ec ∧ Eb ∧(

(σbcdθ)αˆθβ − (d ˆθσbc)β θα

)δ4(x− x) +O(f 4) , (B.18)

O(f 4) =1

2 · 4!

∫W 3

Ec ∧ Eb ∧ Ea(θ)2(ˆθ)2εabcd

(σdαβδ4(x− x)− σe

αβ∂e∂

dδ4(x− x))

+

+i

8

∫W 3

Ec ∧ Eb ∧ (dθσbcσa)β θα (ˆθ)2∂aδ

4(x− x) +

+i

8

∫W 3

Ec ∧ Eb ∧ (d ˆθσbcσa)α

ˆθβ (θ)2∂aδ4(x− x)−

−i∫W 3

Ec ∧ dθγ ∧ d ˆθγσcγγ θαˆθβδ

4(x− x) . (B.19)

109

APPENDIX C

ON ADMISSIBLE VARIATIONS OFSUPERFIELD SUPERGRAVITY

The admissible variations of supervielbein are the variation preserving the superspaceconstraints. In the case of minimal supergravity that read [72, 158]

δEa = Ea(Λ(δ) + Λ(δ))− 1

4Ebσααb [Dα, Dα]δHa + iEαDαδHa − iEαDαδHa , (C.1)

δEα = EaΞαa (δ) + EαΛ(δ) +

1

8EαRσaα

αδHa , (C.2)

where

2Λ(δ) + Λ(δ) = 14σααa DαDαδHa + 1

8GaδH

a + 3(DD − R)δU . (C.3)

The explicit expression for Ξαa (δ) in (C.2), as well as the admissible variations of the spin

connection superform, can be found in [72].

The variation of the closed 4–form (3.22), (3.21) reads [84]

δH4 =1

2Eb ∧ Eα ∧ Eβ ∧ Eγσab (αβDγ)δH

a − 1

2Eb ∧ Eα ∧ Eβ ∧ E γσab αβDγδH

a + c.c.−

− i2Eb ∧ Ea ∧ Eα ∧ Eβ

(σab αβ

(2Λ(δ) + Λ(δ)

)+

1

4σc[a| αβσ|b]

γγ[Dγ, Dγ]δHc

)+ c.c.+

+i

16Eb ∧ Ea ∧ Eα ∧ Eβ(Rσabσc − Rσcσab)αβδH

c+ ∝ Ec ∧ Eb ∧ Ea . (C.4)

The conditions of that δH4 can be expressed in terms of the variation of the 3–form potentialδC3,

δH4 = d(δC3) (C.5)

111

with δC3 decomposed on the basic covariant 3–forms, as in Eq. (3.36), restrict the set ofindependent variations by [84]

(DD − R)δU =1

12(DD − R)

(iδV +

1

2Dαδκα

). (C.6)

This is equivalent to

δU =i

12δV +

1

24Dαδκα +

i

24Dαδνα , (C.7)

where δνα is an additional independent variation (which does not contribute to (DD− R)δUand, hence, to the variations of supergravity potentials).

Factoring out the gauge transformations, we can write the variation δC3, which produces(C.4) through (C.5), in the form (3.36) with [84]

βαβγ(δ) = 0 = βαβγ(δ) , βαβa(δ) = iσaαβδV (C.8)

and

βαβa(δ) = −σab αβ(δHb + σbγγDγδκγ) , (C.9)

βαab(δ) =1

2εabcdσ

cααD

αδHd +1

2σab α

βDβδV −

− i4σab

βγDβDακ

γ +i

4σab α

βDβDβκβ , (C.10)

βabc(δ) =i

8εabcd

((DD − 1

2R

)δHd − c.c.

)+

1

4εabcdG

dδV +

+1

8εabcdσ

dγγ[Dγ, Dγ]δV −i

16εabcdσ

dγγ

((DD +

5

2R

)Dγκγ − c.c.

). (C.11)

The variation of the special minimal supergravity action reads

δSSG =1

6

∫d8ZE

[Ga δH

a + (R− R)iδV]−

− 1

12

∫d8ZE

(RDαδκα + RDαδκα

). (C.12)

Notice that the variations δκα and δκα result in equations DαR = 0 and DαR = 0, whichare satisfied identically due to the minimal (Eq. (3.12)) or special minimal supergravityequations of motion (Eq. (3.40)). In the WZθ=0 gauge (3.49)–(3.53) it is also relativelyeasy to check that δκα does not produce any independent equation for the physical fields ofthe interacting system. This observation has allowed us to simplify the discussion in themain text by neglecting the existence δκα variation.

112

APPENDIX D

SUPERMEMBRANE CURRENTSUPERFIELDS ENTERING THE

SUPERFIELD SUGRA EQUATIONS

The variation of the supermembrane action (3.14) with respect to the vector prepotentialof supergravity, δHa, gives us the vector supercurrent of the form

Ja =

∫W 3

3

EEb ∧ Eα ∧ Eβ σabαβδ

8(Z − Z)−

−∫W 3

3i

E

(∗Ea ∧ Eα +

i

2Eb ∧ Ec ∧ ˆEβεabcdσ

dβα

)Dαδ8(Z − Z) + c.c−

−∫W 3

i

8EEb ∧ Ec ∧ Ed εabcd

(DD − 1

2R

)δ8(Z − Z) + c.c.+

+

∫W 3

1

4E∗ Eb ∧ EbGa δ

8(Z − Z)−

−∫W 3

1

4E∗ Ec ∧ Ebσdαα

(3δcaδ

db − δdaδcb

)[Dα, Dα]δ8(Z − Z) , (D.1)

where E = sdet(EMA(Z)) and

δ8(Z) :=1

16δ4(x) θθ θθ , (D.2)

is the superspace delta function which obeys∫d8Z δ8(Z − Z ′)f(Z) = f(Z ′) for any

superfield f(Z).

113

The supercurrent (D.1) enters the r.h.s. of the vector superfield equation

Ga = T2Ja , (D.3)

which follows from the action (3.14) of the supergravity—supermembrane interacting system.

The scalar superfield equation of the interacting system, which is obtained by varyingthe interacting action (3.43) with respect to the real scalar prepotential of special minimalsupergravity, δS/δV = 0, reads

R− R = −iT2X (D.4)

where

X =6i

E

∫W 3

Ea ∧ Eα ∧ ˆEα σaαα δ8(Z − Z)−

−3

2

∫W 3

Eb ∧ Ea ∧ Eα

Eσabα

βDβδ8(Z − Z) + c.c+

+

∫W 3

Eb ∧ Ec ∧ Ed

8Eεabcdσ

aαα[Dα, Dα]δ8(Z − Z) +

+i

∫W 3

∗Ea ∧ Ea

4E

(DD − R

)δ8(Z − Z) + c.c.+

+

∫W 3

1

4EEb ∧ Ec ∧ EdεabcdG

a δ8(Z − Z) . (D.5)

Notice that, as a consequence of (3.5), the supermembrane current superfields obey

DαJαα = iDαX , DαJαα = −iDαX . (D.6)

In the WZθ=0 gauge (3.49)–(3.53),

iθEα := θαEα

α = θα , iθEα := θαEα

α = θα , (D.7)

θα := θβδ αβ, θα := θβδ α

β, (D.8)

iθEa := θαEα

a = 0 , (D.9)

iθwab := θβwab

β= 0 θα(ξ) = 0 ⇔ θα(ξ) = 0 , ˆθα(ξ) = 0 , (D.10)

these current superfields simply drastically,

Jαα|θ=0 =θβ θβ

8( 3Pab(x)σaαασ

ββb − 2δα

βδαβPbb(x))−

−i(θθ − θθ)32

σaααPa(x)+ ∝ θ∧3 (D.11)

114

Appendix D. Supermembrane current superfields entering the superfield SUGRA equations

and

X|θ=0 = −θσaθ

16Pa + i

(θθ − θθ)16

Paa(x)+ ∝ θ∧3 , (D.12)

where we use the current pre–potential fields defined in (3.62)

Pab(x) :=

∫W 3

1

e∗ ea ∧ eb δ4(x− x) , (D.13)

Pa(x) :=

∫W 3

1

eεabcde

b ∧ ec ∧ ed δ4(x− x) =

= eµa(x)

∫W 3

εµνρσdxν ∧ dxρ ∧ dxσ δ4(x− x) (D.14)

and θ∧3 denotes terms proportional to either θθ θ or θ θθ.

115

APPENDIX E

USEFUL FORMULAE INVOLVING11-DIMENSIONAL AND 9–DIMENSIONAL

GAMMA MATRICES

We use the mostly minus metric convention so that flat spacetime metric readsηab = diag(1,−1, ...,−1). We choose the following SO(1, 1) ⊗ SO(9) invari-ant representation for the 11–dimensional 32 × 32 gamma matrices and charge

conjugation matrix,

(Γa)αβ ≡

(12(Γ# + Γ=),Γi, 1

2(Γ# − Γ=)

), a = 0, 1, . . . , 9, 10 , i = 1, . . . , 9 ,

(Γ#)αβ =

(0 2iδpq0 0

), (Γ=)α

β =

(0 0

−2iδpq 0

), (Γi)α

β =

(−iγipq 0

0 iγipq

), (E.1)

Cαβ = −Cβα =

(0 iδpq−iδpq 0

)= (C−1)αβ =: Cαβ , (E.2)

which are imaginary due to our mostly minus metric convention. In these representationappear the 16×16 9–dimensional Dirac matrices γipq which possesses the following properties

γ(iγj) = δijI16×16, γipq = γiqp := γi(pq),

γi(pqγir)s = δ(pqδr)s. (E.3)

We do not distinguish upper and lower SO(9) spinor indices because the 9–dimensionalcharge conjugation matrix is symmetric allowing us to chose its representation by Kronecherdelta symbol δpq. The matrices δpq,γ

ipq and γijkpq provide a complete basis for the set of

16× 16 symmetric matrices,

δr(qδp)s =1

16δpqδrs +

1

16γipqγ

irs +

1

16 · 4!γijklpq γijklrs . (E.4)

117

In our conventions γ123456789qp = δqp and, consequently,

γi1...i7qp = −1

2εi1...i7jkγjkqp , (E.5)

γi1...i5qp =1

4!εi1...i5j1...j4γj1...j4qp . (E.6)

This, together with (E.1) implies that our 11D dirac matrices obey

Γ0Γ1 . . .Γ9Γ(10) =1

2Γ#Γ=Γ1 . . .Γ9 = −iI32×32 . (E.7)

118

APPENDIX F

MOVING FRAME AND SPINOR MOVINGFRAME VARIABLES

Moving frame and spinor moving frame variables are defined as blocks of, respectively,SO(1, 10) and Spin(1, 10) valued matrices,

U(a)b =

(u=b +u#b

2, uib,

u#b −u=b

2

)∈ SO(1, 10) (F.1)

(i = 1, ..., 9) and

V α(β) =

(v+αq

v−αq

)∈ Spin(1, 10) . (F.2)

We also use

V (β)α =

(vαq

+ , vαq−) ∈ Spin(1, 10) , (F.3)

with

vα−q = iCαβv

−βq , vα

+q = −iCαβv+β

q (F.4)

obeying

V(β)γV (α)

γ = δ(β)(α) =

(δqp 00 δqp

)⇔

v−αq vαp

+ = δqp = v+αq vαp

− ,

v−αq vαp− = 0 = v+α

q vαp+ .

(F.5)

The algebraic properties of moving frame and spinor moving frame variables are summarized

119

as

u=a u

a = = 0 , u=a u

a i = 0 , u =a u

a# = 2 , (F.6)

u#a u

a# = 0 , u #a u

ai = 0 , (F.7)

uiauaj = −δij. (F.8)

v−q Γav−p = u=

a δqp , v+q Γav

+p = u#

a δqp ,

v−q Γav+p = −uiaγiqp , (F.9)

2v−αq v−qβ = Γaαβu=

a , 2v+αq v+

qβ = Γaαβu#

a ,

2v−(αq v+

qβ) = −Γaαβuia . (F.10)

In (F.9) and (F.10) we have used real symmetric 16× 16 9d Dirac matrices γiqp = γipqwhich obey Clifford algebra

γiγj + γjγi = 2δijI16×16 , (F.11)

and

γiq(p1γip2p3) = δq(p1δp2p3) , (F.12)

γijq(q′γip′)p + γijp(q′γ

ip′)q = γjq′p′δqp − δq′p′γ

jqp . (F.13)

Derivatives of the moving frame and spinor moving frame variables are expressed in termsof covariant SO(1,10)

SO(1,1)×SO(9)Cartan forms

Ω=i = u=aduia , Ω#i = u#aduia , (F.14)

and induced SO(1, 1)× SO(9) connection

Ω(0) =1

4u=adu#

a , (F.15)

Ωij = uiaduja . (F.16)

It is convenient to use these latter to define covariant derivative. Then

Dub= := dub

= + 2Ω(0)ub= = ub

iΩ=i , (F.17)

Dub# := dub

# − 2Ω(0)ub# = ub

iΩ#i , (F.18)

Dubi := dub

i − Ωijubj =

1

2ub

#Ω=i +1

2ub

=Ω#i .

(F.19)

120

Appendix F. Moving frame and spinor moving frame variables

Dv−αq := dv−αq + Ω(0)v−αq −1

4Ωijγijqpv

−αp =

= −1

2Ω=iv+α

p γipq , (F.20)

Dv+αq := dv+α

q − Ω(0)v+αq −

1

4Ωijγijqpv

+αp =

= −1

2Ω#iv−αp γipq . (F.21)

The Cartan forms obey

DΩ=i = 0 , DΩ#i = 0 , (F.22)

F (0) := dΩ(0) =1

4Ω= i ∧ Ω# i , (F.23)

Gij := dΩij + Ωik ∧ Ωkj = −Ω= [i ∧ Ω# j] . (F.24)

Notice that, e.g.

DDu#a = 2F (0)u#

a , DDuai = ujaG

ji . (F.25)

The essential variations of moving frame and spinor moving frame variables can be writtenas

δub= = ub

iiδΩ=i , δub

# = ubiiδΩ

#i , (F.26)

δubi =

1

2ub

#iδΩ=i +

1

2ub

=iδΩ#i . (F.27)

δv−αq = −1

2iδΩ

=iv+αp γipq , (F.28)

δv+αq = −1

2iδΩ

#iv−αp γipq , (F.29)

where iδΩ=i and iδΩ

#i are independent variations.The essential variations of the Cartan forms read

δΩ#i = DiδΩ#i , δΩ=i = DiδΩ

=i , (F.30)

δΩij = −Ω=[iiδΩ#j] − Ω#[iiδΩ

=j] , (F.31)

δΩ(0) =1

4Ω=iiδΩ

#i − 1

4Ω#iiδΩ

=i . (F.32)

121

APPENDIX G

EQUATIONS OF MOTION FOR A SINGLEM0–BRANE

In this appendix we collect the equations of motion for the single M0–brane obtainedin chapter 5 from the spinor moving frame action (5.1), (5.2).They read

E= := Eau=a = 0, (G.1)

Ei := Eauia = 0, (G.2)

Dρ# = 0 ⇔ Ω(0) =dρ#

2ρ#, (G.3)

Ω=i = 0 ⇔ Du=a = 0 ⇔ Dv−αq = 0 , (G.4)

E−q := Eαv−qα = 0 . (G.5)

These equations are formulated in terms of pull–backs of bosonic and fermionic supervielbeinforms of flat 11D superspace to the mM0 worldline W 1

Ea = dxa − idθΓaθ , a = 0, 1, ..., 10 , (G.6)

Eα = dθα α = 1, ..., 32 , (G.7)

which are constructed from the coordinate functions xa(τ), θα(τ) of the proper time τ , andof the moving frame and spinor moving frame variables u=

b , uib, v−qα . The properties of these

latter as well as of the Cartan forms Ω=i, Ω(0) and covariant derivatives D are collected inAppendix F.

In (G.6) and in the main text we have used the real symmetric 32× 32 11D Γ–matricesΓaαβ = (γaC)αβ which, together with Γαβa = (Cγa)

αβ, obey Γ(aΓb) = ηabI32×32.

123

APPENDIX H

MULTIPLE M0 EQUATIONS OF MOTION

The mM0 system, which is to say an interacting system of N nearly coincident M0-branes, is described in terms of center of energy variables and the traceless N ×Nmatrices Xi (i = 1, ..., 9), Ψq (q = 1, ..., 16). Our action includes also the auxiliary

N ×N matrix fields: momentum Pi and the 1d SU(N) gauge field Aτ (A = dτAτ ).

The complete list of equations of motion for the mM0 system splits naturally on theequations for the relative motion variables,

DXi = E#Pi + 4iE+q(γiΨ)q,

[Pi,Xi] = 4iΨq , Ψq,

DPi = − 1

16E#[[Xi,Xj]Xj] + 2E# ΨγiΨ + E+qγijqp[Ψp,Xj],

DΨ =i

4E#[Xi, (γiΨ)] +

1

2E+γi Pi − i

16E+γij [Xi,Xj] , (H.1)

and the center of energy equations which can be considered as a deformation of the systemof equations for single M0 brane. After fixing the gauge under a reminiscent of the K9

symmetry, these equation read

E= := Eau=a = 3(ρ#)2tr

(1

2PiDXi +

1

64E#[Xi,Xj]2 − 1

4(E+γijΨ)[Xi,Xj]

), (H.2)

Ei := Eauia = 0 , (H.3)

E−q := Eαv−qα = 0 , (H.4)

Ω=i = 0Ω#i = 0

⇔

Du=

a = 0, Du#a = 0,

Duia = 0 ,Dv−αq = 0 , Dv+α

q = 0 .(H.5)

Dρ# = 0 ⇔ Ω(0) =dρ#

2ρ#, (H.6)

125

As a consequence of the above equation the effective mass M of the mM0 center ofenergy motion,

M2 = 4(ρ#)4H , (H.7)

is a constant

dM2 = 0 . (H.8)

Eq. (H.7) expresses M2 in terms of Lagrange multiplier ρ# and the relative motionHamiltonian (5.74)

H =1

2tr(PiPi

)− 1

64tr[Xi,Xj

]2 − 2 tr(Xi ΨγiΨ

). (H.9)

If we fix the gauge where the composed SO(9) connection and also the SU(N) gaugefield vanish,

Ωij = dτΩijτ = 0 , A = dτAτ = 0, (H.10)

the equations of relative motion and Eq. (H.2) simplify to

∂τ Ψ =i

4e [Xi, (γiΨ)] +

1

2√ρ#E+τ γ

i Pi − i

16√ρ#E+τ γ

ij [Xi, Xj] ,

∂τ

(1

e∂τ Xi

)= − e

16[[Xi, Xj]Xj] + 2 e ΨγiΨ + 4i∂τ

(E+τ γ

iΨ

e√ρ#

)+

1√ρ#E+τ γ

ij[Ψ, Xj] ,

∂τ Xi = ePi +4i√ρ#

(E+τ γ

iΨ), [Pi, Xi] = 4iΨq, Ψq ,

ρ#E=τ = 3tr

(1

2Pi∂τ Xi +

1

64e[Xi, Xj]2 − 1

4√ρ#

(E+τ γ

ijΨ)

[Xi, Xj]

). (H.11)

These equations are written in terms of redefined fields,

Xi = ρ#Xi , Ψq = (ρ#)3/2Ψq ,

Pi = (ρ#)2Pi =1

e

(∂τ Xi − 4i√

ρ#E+τ γ

iΨ

), (H.12)

and

e(τ) = E#τ /ρ

# . (H.13)

126

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